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Triangle T, read by rows, equal to Pascal's triangle to the matrix power of Pascal's triangle, so that T = C^C, where C(n,k) = binomial(n,k) and T(n,k) = A000248(n-k)*C(n,k).
+20
4
1, 1, 1, 3, 2, 1, 10, 9, 3, 1, 41, 40, 18, 4, 1, 196, 205, 100, 30, 5, 1, 1057, 1176, 615, 200, 45, 6, 1, 6322, 7399, 4116, 1435, 350, 63, 7, 1, 41393, 50576, 29596, 10976, 2870, 560, 84, 8, 1, 293608, 372537, 227592, 88788, 24696, 5166, 840, 108, 9, 1
COMMENTS
Column 0 = A000248 (Number of forests with n nodes and height at most 1). Column 1 = A052512 (Number of labeled trees of height 2). Row sums = A080108 (Sum_{k=1..n} k^(n-k) * C(n-1,k-1)). For commuting lower unitriangular matrix A and lower triangular matrix B we define A raised to the matrix power B, denoted by A^B, to be the lower unitriangular matrix Exp(B*Log(A)). Here Exp denotes the matrix exponential defined by the power series
Exp(A) = 1 + A + A^2/2! + A^3/3! + ...
and the matrix logarithm Log(A) is defined by the series
Log(A) = (A-1) - 1/2*(A-1)^2/2 + 1/3*(A-1)^3 - ....
Let A = [f(x),x] and B = [g(x),x] be exponential Riordan arrays in the Appell subgroup and suppose f(0) = 1. Then A and B commute and A^B is the exponential Riordan array [exp(g(x)*log(f(x))),x], also belonging to the Appell group. In the present case we are taking A = B = [exp(x),x], equal to the Pascal triangle A007318. For any lower unitriangular matrix A (with, say, rational entries) the infinite tower of powers A^(A^(A^...))) is well-defined (and also has rational entries). An example is given in the Formula section.
(End)
FORMULA
E.g.f.: exp( x*exp(x) + x*y ).
Exponential Riordan array [exp(x*exp(x)),x] belonging to the Appell group. Thus the e.g.f. for the k-th column of the triangle is x^k/k!*exp(x*exp(x)).
The inverse array, denote it by X, is a signed version of A215652. The infinite tower of matrix powers X^(X^(X^(...))) equals the inverse of Pascal's triangle. (End)
O.g.f.: Sum_{n>=0} x^n / (1 - x*(n+y))^(n+1). - Paul D. Hanna, Aug 03 2014 G.f. for row n: Sum_{k=0..n} binomial(n,k) * (k + y)^(n-k) for n>=0. - Paul D. Hanna, Aug 03 2014 T(n,k) = Sum_{j=0..n-k} C(n,j) * C(n-j,k) * j^(n-k-j) = A000248(n-k)*C(n,k). - Paul D. Hanna, Aug 03 2014
EXAMPLE
E.g.f.: E(x,y) = 1 + (1 + y)*x + (3 + 2*y + y^2)*x^2/2!
+ (10 + 9*y + 3*y^2 + y^3)*x^3/3!
+ (41 + 40*y + 18*y^2 + 4*y^3 + y^4)*x^4/4!
+ (196 + 205*y + 100*y^2 + 30*y^3 + 5*y^4 + y^5)*x^5/5! +...
where E(x,y) = exp(x*y) * exp(x*exp(x)).
O.g.f.: A(x,y) = 1 + (1 + y)*x + (3 + 2*y + y^2)*x^2
+ (10 + 9*y + 3*y^2 + y^3)*x^3
+ (41 + 40*y + 18*y^2 + 4*y^3 + y^4)*x^4
+ (196 + 205*y + 100*y^2 + 30*y^3 + 5*y^4 + y^5)*x^5 +...
where
A(x,y) = 1/(1-x*y) + x/(1-x*(y+1))^2 + x^2/(1-x*(y+2))^3 + x^3/(1-x*(y+3))^4 + x^4/(1-x*(y+4))^5 + x^5/(1-x*(y+5))^6 + x^6/(1-x*(y+6))^7 + x^7/(1-x*(y+7))^8 +...
Triangle begins:
1;
1, 1;
3, 2, 1;
10, 9, 3, 1;
41, 40, 18, 4, 1;
196, 205, 100, 30, 5, 1;
1057, 1176, 615, 200, 45, 6, 1;
6322, 7399, 4116, 1435, 350, 63, 7, 1;
41393, 50576, 29596, 10976, 2870, 560, 84, 8, 1;
293608, 372537, 227592, 88788, 24696, 5166, 840, 108, 9, 1;
2237921, 2936080, 1862685, 758640, 221970, 49392, 8610, 1200, 135, 10, 1; ...
MATHEMATICA
(* The function RiordanArray is defined in A256893. *)
PROG
(PARI) /* By definition C^C: */
{T(n, k)=local(A, C=matrix(n+1, n+1, r, c, binomial(r-1, c-1)), L=matrix(n+1, n+1, r, c, if(r==c+1, c))); A=sum(m=0, n, L^m*C^m/m!); A[n+1, k+1]}
for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")); print(""))
(PARI) /* From e.g.f.: */
{T(n, k)=local(A=1); A=exp( x*y + x*exp(x +x*O(x^n)) ); n!*polcoeff(polcoeff(A, n, x), k, y)}
for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")); print(""))
{T(n, k)=local(A=1); A=sum(k=0, n, x^k/(1 - x*(k+y) +x*O(x^n))^(k+1)); polcoeff(polcoeff(A, n, x), k, y)}
for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")); print(""))
{T(n, k)=local(R); R=sum(k=0, n, (k+y)^(n-k)*binomial(n, k)); polcoeff(R, k, y)}
for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")); print(""))
(PARI) /* From formula for T(n, k) ( Paul D. Hanna, Aug 03 2014): */ {T(n, k) = sum(j=0, n-k, binomial(n, j) * binomial(n-j, k) * j^(n-k-j))}
for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")); print(""))
Euler-Seidel matrix T(k,n) with start sequence A000248, read by antidiagonals.
+20
3
1, 2, 1, 6, 4, 3, 23, 17, 13, 10, 104, 81, 64, 51, 41, 537, 433, 352, 288, 237, 196, 3100, 2563, 2130, 1778, 1490, 1253, 1057, 19693, 16593, 14030, 11900, 10122, 8632, 7379, 6322, 136064, 116371, 99778, 85748, 73848, 63726, 55094, 47715, 41393
COMMENTS
In an Euler-Seidel matrix, the rows are consecutive pairwise sums and the columns consecutive differences, with the first column the inverse binomial transform of the start sequence.
FORMULA
Recurrence: T(0, n) = A000248(n), T(k, n) = T(k-1, n) + T(k-1, n+1).
EXAMPLE
1,1,3,10,41,196,1057,
2,4,13,51,237,1253,7379,
6,17,64,288,1490,8632,55094,
23,81,352,1778,10122,63726,437810,
104,433,2130,11900,73848,501536,3687056,
MATHEMATICA
a248[0] = 1; a248[n_] := Sum[Binomial[n, k]*(n - k)^k, {k, 0, n}];
T[0, n_] := T[0, n] = a248[n];
T[k_, n_] := T[k, n] = T[k - 1, n] + T[k - 1, n + 1];
Floor-Sqrt transform of idempotent endomap numbers ( A000248).
+20
0
1, 1, 1, 3, 6, 14, 32, 79, 203, 541, 1495, 4267, 12543, 37902, 117519, 373236, 1212418, 4023007, 13619993, 46998318, 165144243, 590411905, 2145978422, 7924484313, 29710783080, 113031311232, 436102652463, 1705552475812, 6758097531570, 27119230158894
MATHEMATICA
FSFromExpSeries[f_, x_, n_] := Map[Floor[Sqrt[#]]&, CoefficientList[Series[f, {x, 0, n}], x]Table[k!, {k, 0, n}]]
FSFromExpSeries[Exp[x Exp[x]], x, 40]
a(n) is the number of partitions of n (the partition numbers).
(Formerly M0663 N0244)
+10
3776
1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 101, 135, 176, 231, 297, 385, 490, 627, 792, 1002, 1255, 1575, 1958, 2436, 3010, 3718, 4565, 5604, 6842, 8349, 10143, 12310, 14883, 17977, 21637, 26015, 31185, 37338, 44583, 53174, 63261, 75175, 89134, 105558, 124754, 147273, 173525
COMMENTS
Also number of nonnegative solutions to b + 2c + 3d + 4e + ... = n and the number of nonnegative solutions to 2c + 3d + 4e + ... <= n. - Henry Bottomley, Apr 17 2001 a(n) is also the number of conjugacy classes in the symmetric group S_n (and the number of irreducible representations of S_n).
Also the number of rooted trees with n+1 nodes and height at most 2.
Coincides with the sequence of numbers of nilpotent conjugacy classes in the Lie algebras gl(n). A006950, A015128 and this sequence together cover the nilpotent conjugacy classes in the classical A,B,C,D series of Lie algebras. - Alexander Elashvili, Sep 08 2003 Number of distinct Abelian groups of order p^n, where p is prime (the number is independent of p). - Lekraj Beedassy, Oct 16 2004 Number of graphs on n vertices that do not contain P3 as an induced subgraph. - Washington Bomfim, May 10 2005 Numbers of terms to be added when expanding the n-th derivative of 1/f(x). - Thomas Baruchel, Nov 07 2005 Sequence agrees with expansion of Molien series for symmetric group S_n up to the term in x^n. - Maurice D. Craig (towenaar(AT)optusnet.com.au), Oct 30 2006
Also the number of nonnegative integer solutions to x_1 + x_2 + x_3 + ... + x_n = n such that n >= x_1 >= x_2 >= x_3 >= ... >= x_n >= 0, because by letting y_k = x_k - x_(k+1) >= 0 (where 0 < k < n) we get y_1 + 2y_2 + 3y_3 + ... + (n-1)y_(n-1) + nx_n = n. - Werner Grundlingh (wgrundlingh(AT)gmail.com), Mar 14 2007
Let P(z) := Sum_{j>=0} b_j z^j, b_0 != 0. Then 1/P(z) = Sum_{j>=0} c_j z^j, where the c_j must be computed from the infinite triangular system b_0 c_0 = 1, b_0 c_1 + b_1 c_0 = 0 and so on (Cauchy products of the coefficients set to zero). The n-th partition number arises as the number of terms in the numerator of the expression for c_n: The coefficient c_n of the inverted power series is a fraction with b_0^(n+1) in the denominator and in its numerator having a(n) products of n coefficients b_i each. The partitions may be read off from the indices of the b_i. - Peter C. Heinig (algorithms(AT)gmx.de), Apr 09 2007
a(n) is the number of different ways to run up a staircase with n steps, taking steps of sizes 1, 2, 3, ... and r (r <= n), where the order is not important and there is no restriction on the number or the size of each step taken. - Mohammad K. Azarian, May 21 2008 A sequence of positive integers p = p_1 ... p_k is a descending partition of the positive integer n if p_1 + ... + p_k = n and p_1 >= ... >= p_k. If formally needed p_j = 0 is appended to p for j > k. Let P_n denote the set of these partition for some n >= 1. Then a(n) = 1 + Sum_{p in P_n} floor((p_1-1)/(p_2+1)). (Cf. A000065, where the formula reduces to the sum.) Proof in Kelleher and O'Sullivan (2009). For example a(6) = 1 + 0 + 0 + 0 + 0 + 1 + 0 + 0 + 1 + 1 + 2 + 5 = 11. - Peter Luschny, Oct 24 2010 Let n = Sum( k_(p_m) p_m ) = k_1 + 2k_2 + 5k_5 + 7k_7 + ..., where p_m is the m-th generalized pentagonal number ( A001318). Then a(n) is the sum over all such pentagonal partitions of n of (-1)^(k_5+k_7 + k_22 + ...) ( k_1 + k_2 + k_5 + ...)! /( k_1! k_2! k_5! ...), where the exponent of (-1) is the sum of all the k's corresponding to even-indexed GPN's. - Jerome Malenfant, Feb 14 2011 The matrix of a(n) values
a(0)
a(1) a(0)
a(2) a(1) a(0)
a(3) a(2) a(1) a(0)
....
a(n) a(n-1) a(n-2) ... a(0)
is the inverse of the matrix
1
-1 1
-1 -1 1
0 -1 -1 1
....
-d_n -d_(n-1) -d_(n-2) ... -d_1 1
where d_q = (-1)^(m+1) if q = m(3m-1)/2 = the m-th generalized pentagonal number ( A001318), = 0 otherwise. (End) Let k > 0 be an integer, and let i_1, i_2, ..., i_k be distinct integers such that 1 <= i_1 < i_2 < ... < i_k. Then, equivalently, a(n) equals the number of partitions of N = n + i_1 + i_2 + ... + i_k in which each i_j (1 <= j <= k) appears as a part at least once. To see this, note that the partitions of N of this class must be in 1-to-1 correspondence with the partitions of n, since N - i_1 - i_2 - ... - i_k = n. - L. Edson Jeffery, Apr 16 2011 a(n) is the number of distinct degree sequences over all free trees having n + 2 nodes. Take a partition of the integer n, add 1 to each part and append as many 1's as needed so that the total is 2n + 2. Now we have a degree sequence of a tree with n + 2 nodes. Example: The partition 3 + 2 + 1 = 6 corresponds to the degree sequence {4, 3, 2, 1, 1, 1, 1, 1} of a tree with 8 vertices. - Geoffrey Critzer, Apr 16 2011 a(n) is number of distinct characteristic polynomials among n! of permutations matrices size n X n. - Artur Jasinski, Oct 24 2011 Conjecture: starting with offset 1 represents the numbers of ordered compositions of n using the signed (++--++...) terms of A001318 starting (1, 2, -5, -7, 12, 15, ...). - Gary W. Adamson, Apr 04 2013 (this is true by the pentagonal number theorem, Joerg Arndt, Apr 08 2013) a(n) is also number of terms in expansion of the n-th derivative of log(f(x)). In Mathematica notation: Table[Length[Together[f[x]^n * D[Log[f[x]], {x, n}]]], {n, 1, 20}]. - Vaclav Kotesovec, Jun 21 2013 Conjecture: No a(n) has the form x^m with m > 1 and x > 1. - Zhi-Wei Sun, Dec 02 2013 Partitions of n that contain a part p are the partitions of n - p. Thus, number of partitions of m*n - r that include k*n as a part is A000041(h*n-r), where h = m - k >= 0, n >= 2, 0 <= r < n; see A111295 as an example. - Clark Kimberling, Mar 03 2014 a(n) is the number of compositions of n into positive parts avoiding the pattern [1, 2]. - Bob Selcoe, Jul 08 2014 Conjecture: For any j there exists k such that all primes p <= A000040(j) are factors of one or more a(n) <= a(k). Growth of this coverage is slow and irregular. k = 1067 covers the first 102 primes, thus slower than A000027. - Richard R. Forberg, Dec 08 2014 a(n) is the number of nilpotent conjugacy classes in the order-preserving, order-decreasing and (order-preserving and order-decreasing) injective transformation semigroups. - Ugbene Ifeanyichukwu, Jun 03 2015 Define a segmented partition a(n,k, <s(1)..s(j)>) to be a partition of n with exactly k parts, with s(j) parts t(j) identical to each other and distinct from all the other parts. Note that n >= k, j <= k, 0 <= s(j) <= k, s(1)t(1) + ... + s(j)t(j) = n and s(1) + ... + s(j) = k. Then there are up to a(k) segmented partitions of n with exactly k parts. - Gregory L. Simay, Nov 08 2015 (End)
The polynomials for a(n, k, <s(1), ..., s(j)>) have degree j-1.
a(n, k, <k>) = 1 if n = 0 mod k, = 0 otherwise
a(rn, rk, <r*s(1), ..., r*s(j)>) = a(n, k, <s(1), ..., s(j)>)
a(n odd, k, <all s(j) even>) = 0
Established results can be recast in terms of segmented partitions:
For j(j+1)/2 <= n < (j+1)(j+2)/2, A000009(n) = a(n, 1, <1>) + ... + a(n, j, <j 1's>), j < n a(n, k, <j 1's>) = a(n - j(j-1)/2, k)
(End)
a(10^20) was computed using the NIST Arb package. It has 11140086260 digits and its head and tail sections are 18381765...88091448. See the Johansson 2015 link. - Stanislav Sykora, Feb 01 2016 Satisfies Benford's law [Anderson-Rolen-Stoehr, 2011]. - N. J. A. Sloane, Feb 08 2017 The partition function p(n) is log-concave for all n>25 [DeSalvo-Pak, 2014]. - Michel Marcus, Apr 30 2019 a(n) is also the dimension of the n-th cohomology of the infinite real Grassmannian with coefficients in Z/2. - Luuk Stehouwer, Jun 06 2021 Equivalently, number of idempotent mappings f from a set X of n elements into itself (i.e., satisfying f o f = f) up to permutation (i.e., f~f' :<=> There is a permutation sigma in Sym(X) such that f' o sigma = sigma o f). - Philip Turecek, Apr 17 2023 Conjecture: Each integer n > 2 different from 6 can be written as a sum of finitely many numbers of the form a(k) + 2 (k > 0) with no summand dividing another. This has been verified for n <= 7140. - Zhi-Wei Sun, May 16 2023 a(n) is also the number of partitions of n*(n+3)/2 into n distinct parts. - David García Herrero, Aug 20 2024
REFERENCES
George E. Andrews, The Theory of Partitions, Addison-Wesley, Reading, Mass., 1976.
George E. Andrews and K. Ericksson, Integer Partitions, Cambridge University Press 2004.
T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 307.
R. Ayoub, An Introduction to the Analytic Theory of Numbers, Amer. Math. Soc., 1963; Chapter III.
Mohammad K. Azarian, A Generalization of the Climbing Stairs Problem, Mathematics and Computer Education Journal, Vol. 31, No. 1, pp. 24-28, Winter 1997.
Mohammad K. Azarian, A Generalization of the Climbing Stairs Problem II, Missouri Journal of Mathematical Sciences, Vol. 16, No. 1, Winter 2004, pp. 12-17. Zentralblatt MATH, Zbl 1071.05501.
Bruce C. Berndt, Ramanujan's Notebooks Part V, Springer-Verlag.
B. C. Berndt, Number Theory in the Spirit of Ramanujan, Chap. I Amer. Math. Soc. Providence RI 2006.
Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 999.
J. M. Borwein, D. H. Bailey and R. Girgensohn, Experimentation in Mathematics, A K Peters, Ltd., Natick, MA, 2004. x+357 pp. See p. 183.
L. E. Dickson, History of the Theory of Numbers, Vol.II Chapter III pp. 101-164, Chelsea NY 1992.
N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 37, Eq. (22.13).
H. Gupta et al., Tables of Partitions. Royal Society Mathematical Tables, Vol. 4, Cambridge Univ. Press, 1958, p. 90.
G. H. Hardy and S. Ramanujan, Asymptotic formulas in combinatorial analysis, Proc. London Math. Soc., 17 (1918), 75-.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers (Fifth edition), Oxford Univ. Press (Clarendon), 1979, 273-296.
D. E. Knuth, The Art of Computer Programming, vol. 4A, Combinatorial Algorithms, Section 7.2.1.4, p. 396.
D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, Section XIV.1, p. 491.
S. Ramanujan, Collected Papers, Chap. 25, Cambridge Univ. Press 1927 (Proceedings of the Camb. Phil. Soc., 19 (1919), pp. 207-213).
S. Ramanujan, Collected Papers, Chap. 28, Cambridge Univ. Press 1927 (Proceedings of the London Math. Soc., 2, 18(1920)).
S. Ramanujan, Collected Papers, Chap. 30, Cambridge Univ. Press 1927 (Mathematische Zeitschrift, 9 (1921), pp. 147-163).
S. Ramanujan, Collected Papers, Ed. G. H. Hardy et al., Cambridge 1927; Chelsea, NY, 1962. See Table IV on page 308.
J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 122.
J. E. Roberts, Lure of the Integers, pp. 168-9 MAA 1992.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
R. E. Tapscott and D. Marcovich, "Enumeration of Permutational Isomers: The Porphyrins", Journal of Chemical Education, 55 (1978), 446-447.
Robert M. Young, "Excursions in Calculus", Mathematical Association of America, p. 367.
LINKS
Milton Abramowitz and Irene A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972, p. 836. [scanned copy] James Grime and Brady Haran, Partitions, Numberphile video (2016). Johannes W. Meijer, Euler's ship on the Pentagonal Sea, pdf and jpg.
FORMULA
G.f.: Product_{k>0} 1/(1-x^k) = Sum_{k>= 0} x^k Product_{i = 1..k} 1/(1-x^i) = 1 + Sum_{k>0} x^(k^2)/(Product_{i = 1..k} (1-x^i))^2.
G.f.: 1 + Sum_{n>=1} x^n/(Product_{k>=n} 1-x^k). - Joerg Arndt, Jan 29 2011 a(n) - a(n-1) - a(n-2) + a(n-5) + a(n-7) - a(n-12) - a(n-15) + ... = 0, where the sum is over n-k and k is a generalized pentagonal number ( A001318) <= n and the sign of the k-th term is (-1)^([(k+1)/2]). See A001318 for a good way to remember this! a(n) = (1/n) * Sum_{k=0..n-1} sigma(n-k)*a(k), where sigma(k) is the sum of divisors of k ( A000203). a(n) ~ 1/(4*n*sqrt(3)) * e^(Pi * sqrt(2n/3)) as n -> infinity (Hardy and Ramanujan). See A050811. a(n) = a(0)*b(n) + a(1)*b(n-2) + a(2)*b(n-4) + ... where b = A000009. It appears that the above approximation from Hardy and Ramanujan can be refined as
a(n) ~ 1/(4*n*sqrt(3)) * e^(Pi * sqrt(2n/3 + c0 + c1/n^(1/2) + c2/n + c3/n^(3/2) + c4/n^2 + ...)), where the coefficients c0 through c4 are approximately
c0 = -0.230420145062453320665537
c1 = -0.0178416569128570889793
c2 = 0.0051329911273
c3 = -0.0011129404
c4 = 0.0009573,
as n -> infinity. (End)
c0 = -0.230420145062453320665536704197233... = -1/36 - 2/Pi^2
c1 = -0.017841656912857088979502135349949... = 1/(6*sqrt(6)*Pi) - sqrt(3/2)/Pi^3
c2 = 0.005132991127342167594576391633559... = 1/(2*Pi^4)
c3 = -0.001112940489559760908236602843497... = 3*sqrt(3/2)/(4*Pi^5) - 5/(16*sqrt(6)*Pi^3)
c4 = 0.000957343284806972958968694349196... = 1/(576*Pi^2) - 1/(24*Pi^4) + 93/(80*Pi^6)
a(n) ~ exp(Pi*sqrt(2*n/3))/(4*sqrt(3)*n) * (1 - (sqrt(3/2)/Pi + Pi/(24*sqrt(6)))/sqrt(n) + (1/16 + Pi^2/6912)/n).
a(n) ~ exp(Pi*sqrt(2*n/3) - (sqrt(3/2)/Pi + Pi/(24*sqrt(6)))/sqrt(n) + (1/24 - 3/(4*Pi^2))/n) / (4*sqrt(3)*n).
(End)
a(n) < exp( (2/3)^(1/2) Pi sqrt(n) ) (Ayoub, p. 197).
a(n) = Sum_{i=0..n-1} P(i, n-i), where P(x, y) is the number of partitions of x into at most y parts and P(0, y)=1. - Jon Perry, Jun 16 2003 G.f.: Product_{i>=1} Product_{j>=0} (1+x^((2i-1)*2^j))^(j+1). - Jon Perry, Jun 06 2004 a(n) = determinant of the n X n Toeplitz matrix:
1 -1
1 1 -1
0 1 1 -1
0 0 1 1 -1
-1 0 0 1 1 -1
. . .
d_n d_(n-1) d_(n-2)...1
where d_q = (-1)^(m+1) if q = m(3m-1)/2 = p_m, the m-th generalized pentagonal number ( A001318), otherwise d_q = 0. Note that the 1's run along the diagonal and the -1's are on the superdiagonal. The (n-1) row (not written) would end with ... 1 -1. (End) Empirical: let F*(x) = Sum_{n=0..infinity} p(n)*exp(-Pi*x*(n+1)), then F*(2/5) = 1/sqrt(5) to a precision of 13 digits.
F*(4/5) = 1/2+3/2/sqrt(5)-sqrt(1/2*(1+3/sqrt(5))) to a precision of 28 digits. These are the only values found for a/b when a/b is from F60, Farey fractions up to 60. The number for F*(4/5) is one of the real roots of 25*x^4 - 50*x^3 - 10*x^2 - 10*x + 1. Note here the exponent (n+1) compared to the standard notation with n starting at 0. - Simon Plouffe, Feb 23 2011 The constant (2^(7/8)*GAMMA(3/4))/(exp(Pi/6)*Pi^(1/4)) = 1.0000034873... when expanded in base exp(4*Pi) will give the first 52 terms of a(n), n>0, the precision needed is 300 decimal digits. - Simon Plouffe, Mar 02 2011 G.f.: A(x)=1+x/(G(0)-x); G(k) = 1 + x - x^(k+1) - x*(1-x^(k+1))/G(k+1); (continued fraction Euler's kind, 1-step ). - Sergei N. Gladkovskii, Jan 25 2012 G.f.: 1 + x*(1 - G(0))/(1-x) where G(k) = 1 - 1/(1-x^(k+1))/(1-x/(x-1/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jan 22 2013 G.f.: Q(0) where Q(k) = 1 + x^(4*k+1)/( (x^(2*k+1)-1)^2 - x^(4*k+3)*(x^(2*k+1)-1)^2/( x^(4*k+3) + (x^(2*k+2)-1)^2/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Feb 16 2013 a(n-1) = Sum_{parts k in all partitions of n} mu(k), where mu(k) is the arithmetical Möbius function (see A008683). Let P(2,n) denote the set of partitions of n into parts k >= 2. Then a(n-2) = -Sum_{parts k in all partitions in P(2,n)} mu(k).
n*( a(n) - a(n-1) ) = Sum_{parts k in all partitions in P(2,n)} k (see A138880). Let P(3,n) denote the set of partitions of n into parts k >= 3. Then
a(n-3) = (1/2)*Sum_{parts k in all partitions in P(3,n)} phi(k), where phi(k) is the Euler totient function (see A000010). Using this result and Mertens's theorem on the average order of the phi function, we can find an approximate 3-term recurrence for the partition function: a(n) ~ a(n-1) + a(n-2) + (Pi^2/(3*n) - 1)*a(n-3). For example, substituting the values a(47) = 124754, a(48) = 147273 and a(49) = 173525 into the recurrence gives the approximation a(50) ~ 204252.48... compared with the true value a(50) = 204226. (End) A production matrix for the sequence with offset 1 is M, an infinite n x n matrix of the following form:
a, 1, 0, 0, 0, 0, ...
b, 0, 1, 0, 0, 0, ...
c, 0, 0, 1, 0, 0, ...
d, 0, 0, 0, 1, 0, ...
.
.
... such that (a, b, c, d, ...) is the signed version of A080995 with offset 1: (1,1,0,0,-1,0,-1,...) and a(n) is the upper left term of M^n.
This operation is equivalent to the g.f. (1 + x + 2x^2 + 3x^3 + 5x^4 + ...) = 1/(1 - x - x^2 + x^5 + x^7 - x^12 - x^15 + x^22 + ...). (End)
a(n) = Sum_{k=-inf..+inf} (-1)^k a(n-k(3k-1)/2) with a(0)=1 and a(negative)=0. The sum can be restricted to the (finite) range from k = (1-sqrt(1-24n))/6 to (1+sqrt(1-24n))/6, since all terms outside this range are zero. - Jos Koot, Jun 01 2016 G.f.: (conjecture) (r(x) * r(x^2) * r(x^4) * r(x^8) * ...) where r(x) is A000009: (1, 1, 1, 2, 2, 3, 4, ...). - Gary W. Adamson, Sep 18 2016; Doron Zeilberger observed today that "This follows immediately from Euler's formula 1/(1-z) = (1+z)*(1+z^2)*(1+z^4)*(1+z^8)*..." Gary W. Adamson, Sep 20 2016 a(n) ~ 2*Pi * BesselI(3/2, sqrt(24*n-1)*Pi/6) / (24*n-1)^(3/4). - Vaclav Kotesovec, Jan 11 2017 G.f.: Product_{k>=1} (1 + x^k)/(1 - x^(2*k)). - Ilya Gutkovskiy, Jan 23 2018 a(n) = p(1, n) where p(k, n) = p(k+1, n) + p(k, n-k) if k < n, 1 if k = n, and 0 if k > n. p(k, n) is the number of partitions of n into parts >= k. - Lorraine Lee, Jan 28 2020 Sum_{n>=0} a(n)/exp(Pi*n) = 2^(3/8)*Gamma(3/4)/(Pi^(1/4)*exp(Pi/24)).
Sum_{n>=0} a(n)/exp(2*Pi*n) = 2^(1/2)*Gamma(3/4)/(Pi^(1/4)*exp(Pi/12)).
[These are the reciprocals of phi(exp(-Pi)) ( A259148) and phi(exp(-2*Pi)) ( A259149), where phi(q) is the Euler modular function. See B. C. Berndt (RLN, Vol. V, p. 326), and formulas (13) and (14) in I. Mező, 2013. - Peter Luschny, Mar 13 2021] a(n) = A008284(2*n,n) is also the number of partitions of 2n into n parts. - Ryan Brooks, Jun 11 2022 a(n) ~ exp(Pi*sqrt(2*n/3)) / (4*n*sqrt(3)) * (1 + Sum_{r>=1} w(r)/n^(r/2)), where w(r) = 1/(-4*sqrt(6))^r * Sum_{k=0..(r+1)/2} binomial(r+1,k) * (r+1-k) / (r+1-2*k)! * (Pi/6)^(r-2*k) [Cormac O'Sullivan, 2023, pp. 2-3]. - Vaclav Kotesovec, Mar 15 2023
EXAMPLE
a(5) = 7 because there are seven partitions of 5, namely: {1, 1, 1, 1, 1}, {2, 1, 1, 1}, {2, 2, 1}, {3, 1, 1}, {3, 2}, {4, 1}, {5}. - Bob Selcoe, Jul 08 2014 G.f. = 1 + x + 2*x^2 + 3*x^3 + 5*x^4 + 7*x^5 + 11*x^6 + 15*x^7 + 22*x^8 + ...
G.f. = 1/q + q^23 + 2*q^47 + 3*q^71 + 5*q^95 + 7*q^119 + 11*q^143 + 15*q^167 + ...
There are up to a(4)=5 segmented partitions of the partitions of n with exactly 4 parts. They are a(n,4, <4>), a(n,4,<3,1>), a(n,4,<2,2>), a(n,4,<2,1,1>), a(n,4,<1,1,1,1>).
The partition 8,8,8,8 is counted in a(32,4,<4>).
The partition 9,9,9,5 is counted in a(32,4,<3,1>).
The partition 11,11,5,5 is counted in a(32,4,<2,2>).
The partition 13,13,5,1 is counted in a(32,4,<2,1,1>).
The partition 14,9,6,3 is counted in a(32,4,<1,1,1,1>).
a(n odd,4,<2,2>) = 0.
a(12, 6, <2,2,2>) = a(6,3,<1,1,1>) = a(6-3,3) = a(3,3) = 1. The lone partition is 3,3,2,2,1,1.
(End)
MAPLE
A000041 := n -> combinat:-numbpart(n): [seq( A000041(n), n=0..50)]; # Warning: Maple 10 and 11 give incorrect answers in some cases: A110375.
spec := [B, {B=Set(Set(Z, card>=1))}, unlabeled ];
[seq(combstruct[count](spec, size=n), n=0..50)];
with(combstruct):ZL0:=[S, {S=Set(Cycle(Z, card>0))}, unlabeled]: seq(count(ZL0, size=n), n=0..45); # Zerinvary Lajos, Sep 24 2007 G:={P=Set(Set(Atom, card>0))}: combstruct[gfsolve](G, labeled, x); seq(combstruct[count]([P, G, unlabeled], size=i), i=0..45); # Zerinvary Lajos, Dec 16 2007 # Using the function EULER from Transforms (see link at the bottom of the page).
MATHEMATICA
Table[ PartitionsP[n], {n, 0, 45}]
a[ n_] := SeriesCoefficient[ q^(1/24) / DedekindEta[ Log[q] / (2 Pi I)], {q, 0, n}]; (* Michael Somos, Jul 11 2011 *) a[ n_] := SeriesCoefficient[ 1 / Product[ 1 - x^k, {k, n}], {x, 0, n}]; (* Michael Somos, Jul 11 2011 *) a[0] := 1; a[n_] := a[n] = Block[{k=1, s=0, i=n-1}, While[i >= 0, s=s-(-1)^k (a[i]+a[i-k]); k=k+1; i=i-(3 k-2)]; s]; Map[a, Range[0, 49]] (* Oliver Seipel, Jun 01 2024 after Euler *)
PROG
(Magma) a:= func< n | NumberOfPartitions(n) >; [ a(n) : n in [0..10]];
(PARI) {a(n) = if( n<0, 0, polcoeff( 1 / eta(x + x * O(x^n)), n))};
(PARI) /* The Hardy-Ramanujan-Rademacher exact formula in PARI is as follows (this is no longer necessary since it is now built in to the numbpart command): */
Psi(n, q) = local(a, b, c); a=sqrt(2/3)*Pi/q; b=n-1/24; c=sqrt(b); (sqrt(q)/(2*sqrt(2)*b*Pi))*(a*cosh(a*c)-(sinh(a*c)/c))
L(n, q) = if(q==1, 1, sum(h=1, q-1, if(gcd(h, q)>1, 0, cos((g(h, q)-2*h*n)*Pi/q))))
g(h, q) = if(q<3, 0, sum(k=1, q-1, k*(frac(h*k/q)-1/2)))
part(n) = round(sum(q=1, max(5, 0.5*sqrt(n)), L(n, q)*Psi(n, q)))
(PARI) {a(n) = numbpart(n)};
(PARI) {a(n) = if( n<0, 0, polcoeff( sum( k=1, sqrtint(n), x^k^2 / prod( i=1, k, 1 - x^i, 1 + x * O(x^n))^2, 1), n))};
(PARI) f(n)= my(v, i, k, s, t); v=vector(n, k, 0); v[n]=2; t=0; while(v[1]<n, i=2; while(v[i]==0, i++); v[i]--; s=sum(k=i, n, k*v[k]); while(i>1, i--; s+=i*(v[i]=(n-s)\i)); t++); t \\ Thomas Baruchel, Nov 07 2005 (PARI) a(n)=if(n<0, 0, polcoeff(exp(sum(k=1, n, x^k/(1-x^k)/k, x*O(x^n))), n)) \\ Joerg Arndt, Apr 16 2010 (MuPAD) combinat::partitions::count(i) $i=0..54 // Zerinvary Lajos, Apr 16 2007 (Sage) [number_of_partitions(n) for n in range(46)] # Zerinvary Lajos, May 24 2009 (Sage)
@CachedFunction
if n == 0: return 1
S = 0; J = n-1; k = 2
while 0 <= J:
S = S+T if is_odd(k//2) else S-T
J -= k if is_odd(k) else k//2
k += 1
return S
(Sage) # uses[EulerTransform from A166861] a = BinaryRecurrenceSequence(1, 0)
b = EulerTransform(a)
(Haskell)
import Data.MemoCombinators (memo2, integral)
a000041 n = a000041_list !! n
a000041_list = map (p' 1) [0..] where
p' = memo2 integral integral p
p _ 0 = 1
p k m = if m < k then 0 else p' k (m - k) + p' (k + 1) m
(GAP) List([1..10], n->Size(OrbitsDomain(SymmetricGroup(IsPermGroup, n), SymmetricGroup(IsPermGroup, n), \^))); # Attila Egri-Nagy, Aug 15 2014 (Perl) use ntheory ":all"; my @p = map { partitions($_) } 0..100; say "[@p]"; # Dana Jacobsen, Sep 06 2015 (Racket)
#lang racket
; SUM(k, -inf, +inf) (-1)^k p(n-k(3k-1)/2)
; For k outside the range (1-(sqrt(1-24n))/6 to (1+sqrt(1-24n))/6) argument n-k(3k-1)/2 < 0.
; Therefore the loops below are finite. The hash avoids repeated identical computations.
(define (p n) ; Nr of partitions of n.
(hash-ref h n
(λ ()
(define r
(+
(let loop ((k 1) (n (sub1 n)) (s 0))
(if (< n 0) s
(loop (add1 k) (- n (* 3 k) 1) (if (odd? k) (+ s (p n)) (- s (p n))))))
(let loop ((k -1) (n (- n 2)) (s 0))
(if (< n 0) s
(loop (sub1 k) (+ n (* 3 k) -2) (if (odd? k) (+ s (p n)) (- s (p n))))))))
(hash-set! h n r)
r)))
(define h (make-hash '((0 . 1))))
; (for ((k (in-range 0 50))) (printf "~s, " (p k))) runs in a moment.
(Python)
from sympy.ntheory import npartitions
print([npartitions(i) for i in range(101)]) # Indranil Ghosh, Mar 17 2017 (Julia) # DedekindEta is defined in A000594A000041List(len) = DedekindEta(len, -1)
CROSSREFS
Cf. A000009, A000079, A000203, A001318, A008284, A026820, A065446, A078506, A113685, A132311, A000248.
KEYWORD
core,easy,nonn,nice,changed
EXTENSIONS
Additional comments from Ola Veshta (olaveshta(AT)my-deja.com), Feb 28 2001
Additional comments from Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), Apr 07 2001
n-th derivative of x^x at x=1. Also called Lehmer-Comtet numbers.
(Formerly M0868)
+10
41
1, 1, 2, 3, 8, 10, 54, -42, 944, -5112, 47160, -419760, 4297512, -47607144, 575023344, -7500202920, 105180931200, -1578296510400, 25238664189504, -428528786243904, 7700297625889920, -146004847062359040, 2913398154375730560, -61031188196889482880
REFERENCES
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 139, table at foot of page.
G. H. Hardy, A Course of Pure Mathematics, 10th ed., Cambridge University Press, 1960, p. 428.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
FORMULA
For n>0, a(n) = Sum_{k=0..n} b(n, k), where b(n, k) is a Lehmer-Comtet number of the first kind (see A008296). a(0) = 1, a(n) = a(n-1)+Sum_{k=0..n-2} (-1)^(n-k)*(n-2-k)!*binomial(n-1, k)*a(k).
a(n) = Sum_{k=0..n} (-1)^(n-k)* A293297(k)*binomial(n, k). a(n) = Sum_{k=0..n} (-1)^k* A203852(k)*binomial(n, k). (End)
MAPLE
A005727 := proc(n) option remember; `if`(n=0, 1, A005727(n-1)+add((-1)^(n-k)*(n-2-k)!*binomial(n-1, k)* A005727(k), k=0..n-2)) end:
MATHEMATICA
NestList[ Factor[ D[ #1, x ] ]&, x^x, n ] /. (x->1)
Range[0, 22]! CoefficientList[ Series[(1 + x)^(1 + x), {x, 0, 22}], x] (* Robert G. Wilson v, Feb 03 2013 *)
PROG
(PARI) a(n)=if(n<0, 0, n!*polcoeff((1+x+x*O(x^n))^(1+x), n))
Number of forests with n nodes and height at most 2.
(Formerly M3021 N1223)
+10
33
1, 1, 3, 16, 101, 756, 6607, 65794, 733833, 9046648, 121961051, 1782690174, 28055070397, 472594822324, 8479144213191, 161340195463066, 3243707386310033, 68679247688467056, 1526976223741111987, 35557878951515668726, 865217354118762606021
COMMENTS
Equivalently, the number of mappings from a set of n elements into itself where f(f(x)) = f(f(f(x))). - Chad Brewbaker, Mar 26 2014
REFERENCES
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
FORMULA
E.g.f.: exp(x*exp(x*exp(x))).
a(n) = n!*sum(m=1..n-1, sum(k=1..n-m, (k^(n-m-k)*m^k)/(k!*(n-m-k)!))/m!)+1. - Vladimir Kruchinin, May 28 2011
EXAMPLE
G.f. = 1 + x + 3*x^2 + 16*x^3 + 101*x^4 + 756*x^5 + 6607*x^6 + 65794*x^7 + ... - Michael Somos, Jul 03 2018
MATHEMATICA
nn = 20; Range[0, nn]! CoefficientList[Series[Exp[x*Exp[x*Exp[x]]], {x, 0, nn}], x] (* T. D. Noe, Jun 21 2012 *) a[ n_] := If[ n < 0, 0, 1 + n! Sum[ Sum[ k^(n - m - k) m^k / (k! (n - m - k)!), {k, n - m}] / m!, {m, n - 1}]]; (* Michael Somos, Jul 03 2018 *)
PROG
(Maxima) a(n):=n!*sum(sum((k^(n-m-k)*m^k)/(k!*(n-m-k)!), k, 1, n-m)/m!, m, 1, n-1)+1; /* Vladimir Kruchinin, May 28 2011 */ (PARI) x='x+O('x^66); Vec(serlaplace(exp(x*exp(x*exp(x))))) /* show terms with a(0)=1 */ /* Joerg Arndt, May 28 2011 */
Expansion of e.g.f.: exp(-x - x^2/2 + x*exp(x)).
+10
32
1, 0, 1, 3, 7, 35, 171, 847, 5041, 32643, 223705, 1659581, 13182159, 110802133, 984241363, 9212696235, 90477239521, 929604133343, 9969157068273, 111329454692485, 1291932988047775, 15550838026589061, 193833398512358011, 2498039016973836491
COMMENTS
The number of simple labeled graphs on n nodes whose connected components are stars. - Geoffrey Critzer, Dec 10 2011 Equivalently, the number of minimal edge covers of the complete graph K_n. - Andrew Howroyd, Aug 04 2017
REFERENCES
R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.15(b).
FORMULA
a(n) = n!*Sum_{k=1..n} (1/k!)*( binomial(k, n-k)*2^(k-n)*(-1)^k + Sum_{j=1..k} binomial(k,j)* (Sum_{i=j..n-k+j} (j^(i-j)/(i-j)! * binomial(k-j,n-i-k+j)*(1/2)^(n-i-k+j)*(-1)^(k-j)) ) ), n>0. - Vladimir Kruchinin, Sep 10 2010 a(n) ~ n^n / (exp(r^2/2 + n*r/(1+r)) * r^n * sqrt(r^2*(1+r)/n + 2+r-1/(1+r))), where r is the root of the equation r*(exp(r)*(1+r)-1-r) = n.
(a(n)/n!)^(1/n) ~ exp(1/(2*LambertW(sqrt(n)/2)))/(2*LambertW(sqrt(n)/2)).
(End)
MATHEMATICA
nn = 30; a = x Exp[x]; Range[0, nn]! CoefficientList[Series[Exp[a - x^2/2! - x], {x, 0, nn}], x] (* Geoffrey Critzer, Dec 10 2011 *) CoefficientList[Series[Exp[-x - x^2/2 + x Exp[x]], {x, 0, 30}], x] Range[0, 30]! (* Eric W. Weisstein, Aug 10 2017 *) Table[n! Sum[1/k! (Binomial[k, n-k] 2^(k-n) (-1)^k + Sum[Binomial[k, j] Sum[j^(i-j)/(i-j)! Binomial[k-j, n-i-k+j] 2^(i-j+k-n) (-1)^(k-j), {i, j, n-k+j}], {j, k}]), {k, n}], {n, 30}] (* Eric W. Weisstein, Aug 10 2017 *)
PROG
(Maxima) a(n):=n!*sum((binomial(k, n-k)*2^(k-n)*(-1)^k +sum(binomial(k, j) *sum(j^(i-j)/(i-j)!*binomial(k-j, n-i-k+j)*(1/2)^(n-i-k+j)*(-1)^(k-j), i, j, n-k+j), j, 1, k))/k!, k, 1, n); /* Vladimir Kruchinin, Sep 10 2010 */ (PARI) x='x+O('x^30); Vec(serlaplace(exp(-x-1/2*x^2+x*exp(x)))) \\ Altug Alkan, Aug 10 2017 (Magma) m:=30; R<x>:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( Exp(-x -x^2/2 +x*Exp(x)) )); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, May 15 2019 (Sage) m = 30; T = taylor(exp(-x -x^2/2 +x*exp(x)), x, 0, m); [factorial(n)*T.coefficient(x, n) for n in (0..m)] # G. C. Greubel, May 15 2019
Expansion of e.g.f. exp(x*exp(x)-x).
+10
25
1, 0, 2, 3, 16, 65, 336, 1897, 11824, 80145, 586000, 4588001, 38239224, 337611001, 3144297352, 30779387745, 315689119456, 3383159052833, 37790736663456, 439036039824193, 5294386116882280, 66155074120062921, 855156188538926296, 11418964004032623809
COMMENTS
a(n) is the number of forests of rooted labeled trees with height exactly one. Equivalently, the number of idempotent mappings from {1,2,...,n} into {1,2,...,n} where each fixed point has at least one (other than itself) element mapped to it. See the second summation formula provided by Vladeta Jovovic with conditions on k, the number of fixed points. - Geoffrey Critzer, Sep 20 2012
FORMULA
a(n) = Sum_{k=0..n} binomial(n, k)*(n-k-1)^k. - Vladeta Jovovic, Apr 12 2003 a(n) = Sum_{k=0..floor(n/2)} binomial(n, k)*k!*Stirling2(n-k, k). - Vladeta Jovovic, Dec 19 2004 a(n) ~ exp((1-r*(n+r))/(1+r)) * n^(n+1/2) * sqrt(1+r) / (r^n * sqrt((1+r)^3 + n*(1+3*r+r^2))), where r satisfies exp(r)*(1+r) - (1+n)/r = 1. - Vaclav Kotesovec, Aug 04 2014 (a(n)/n!)^(1/n) ~ exp(1/(2*LambertW(sqrt(n)/2))) / (2*LambertW(sqrt(n)/2)). - Vaclav Kotesovec, Aug 06 2014
MAPLE
spec := [S, {S=Set(Tree), Tree=Prod(Z, Set(Z, 0 < card))}, labeled]: seq(combstruct[count](spec, size=n), n=0..20);
MATHEMATICA
nn=20; Range[0, nn]! CoefficientList[Series[Exp[x(Exp[x]-1)], {x, 0, nn}], x] (* Geoffrey Critzer, Sep 20 2012 *)
PROG
(PARI) my(x='x+O('x^30)); Vec(serlaplace( exp(x*exp(x)-x) )) \\ G. C. Greubel, Nov 15 2017 (PARI) my(N=30, x='x+O('x^N)); Vec(sum(k=0, N, x^k/(1-(k-1)*x)^(k+1))) \\ Seiichi Manyama, Aug 29 2022 (Magma) m:=30; R<x>:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( Exp(x*Exp(x)-x) )); [Factorial(n-1)*b[n]: n in [1..m-1]]; // G. C. Greubel, May 13 2019 (Sage) m = 30; T = taylor(exp(x*exp(x)-x), x, 0, m); [factorial(n)*T.coefficient(x, n) for n in (0..m)] # G. C. Greubel, May 13 2019
AUTHOR
encyclopedia(AT)pommard.inria.fr, Jan 25 2000
Expansion of e.g.f. exp( x * exp(x^2) ).
+10
23
1, 1, 1, 7, 25, 121, 841, 4831, 40657, 325585, 2913841, 29910871, 301088041, 3532945417, 41595396025, 531109561711, 7197739614241, 100211165640481, 1507837436365537, 23123578483200295, 376697477235716281, 6348741961892933401, 111057167658053740201, 2032230051717594032767
FORMULA
a(n) = n!*Sum_{m=floor((n+1)/2)..n} (2*m-n)^(n-m)/((2*m-n)!*(n-m)!). - Vladimir Kruchinin, Mar 09 2013 a(n) ~ n^n / (r^n * exp((2*r^2*n)/(1+2*r^2)) * sqrt(3+2*r^2 - 2/(1 + 2*r^2))), where r is the root of the equation r*exp(r^2)*(1+2*r^2) = n.
(a(n)/n!)^(1/n) ~ exp(1/(3*LambertW(2^(1/3)*n^(2/3)/3))) * sqrt(2/(3*LambertW(2^(1/3)*n^(2/3)/3))).
(End)
MATHEMATICA
With[{nn = 25}, CoefficientList[Series[Exp[x Exp[x^2]], {x, 0, nn}], x] Range[0, nn]!] (* Bruno Berselli, Sep 14 2012 *)
PROG
(PARI)
x='x+O('x^66);
Vec(serlaplace(exp( x * exp(x^2) )))
(PARI) a(n) = n!*sum(k=0, n\2, (n-2*k)^k/(k!*(n-2*k)!)); \\ Seiichi Manyama, Aug 18 2022
CROSSREFS
Cf. A216507 (e.g.f. exp(x^2*exp(x))), A216689 (e.g.f. exp(x*exp(x)^2)).
Triangle of idempotent numbers binomial(n,k)*k^(n-k), version 1.
+10
21
1, 0, 1, 0, 2, 1, 0, 3, 6, 1, 0, 4, 24, 12, 1, 0, 5, 80, 90, 20, 1, 0, 6, 240, 540, 240, 30, 1, 0, 7, 672, 2835, 2240, 525, 42, 1, 0, 8, 1792, 13608, 17920, 7000, 1008, 56, 1, 0, 9, 4608, 61236, 129024, 78750, 18144, 1764, 72, 1, 0, 10, 11520, 262440
COMMENTS
T(n,k) = C(n,k)*k^(n-k) is the number of functions f from domain [n] to codomain [n+1] such that f(x)=n+1 for exactly k elements x of [n] and f(f(x))=n+1 for the remaining n-k elements x of [n]. Subsequently, row sums of T(n,k) provide the number of functions f:[n]->[n+1] such that either f(x)=n+1 or f(f(x))=n+1 for every x in [n]. We note that there are C(n,k) ways to choose the k elements mapped to n+1 and there are k^(n-k) ways to map n-k elements to a set of k elements. - Dennis P. Walsh, Sep 05 2012 The above conjecture is correct. This triangle is the exponential Riordan array [1, x*exp(x)]. Thus the inverse array is the exponential Riordan array [ 1, W(x)], which equals A137452. - Peter Bala, Apr 08 2013
REFERENCES
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 91, #43 and p. 135, [3i'].
FORMULA
Up to signs, this is the triangle of connection constants expressing the monomials x^n as a linear combination of the Abel polynomials A(k,x) := x*(x+k)^(k-1), 0 <= k <= n. O.g.f. for the k-th column: A(-k,1/x) = x^k/(1-k*x)^(k+1). Cf. A061356. Examples are given below. - Peter Bala, Oct 09 2011 The o.g.f.'s for the diagonals of this triangle are the rational functions occurring in the expansion of the compositional inverse (with respect to x) (x-t*x*exp(x))^-1 = x/(1-t) + 2*t/(1-t)^3*x^2/2! + (3*t+9*t^2)/(1-t)^5*x^3/3! + (4*t+52*t^2+64*t^3)/(1-t)^7*x^4/4! + .... For example, the o.g.f. for second subdiagonal is (3*t+9*t^2)/(1-t)^5 = 3*t + 24*t^2 + 90*t^3 + 240*t^4 + .... See the Bala link. The coefficients of the numerator polynomials are listed in A202017. - Peter Bala, Dec 08 2011 Recurrence equation: T(n+1,k+1) = Sum_{j=0..n-k} (j+1)*binomial(n,j)*T(n-j,k). - Peter Bala, Jan 13 2015
EXAMPLE
Triangle begins:
1;
0, 1;
0, 2, 1;
0, 3, 6, 1;
0, 4, 24, 12, 1;
0, 5, 80, 90, 20, 1;
0, 6, 240, 540, 240, 30, 1;
0, 7, 672, 2835, 2240, 525, 42, 1;
Row 4. Expansion of x^4 in terms of Abel polynomials:
x^4 = -4*x+24*x*(x+2)-12*x*(x+3)^2+x*(x+4)^3.
O.g.f. for column 2: A(-2,1/x) = x^2/(1-2*x)^3 = x^2+6*x^3+24*x^4+80*x^5+....
MAPLE
T:= (n, k)-> binomial(n, k) *k^(n-k):
MATHEMATICA
nn=10; f[list_]:=Select[list, #>0&]; Prepend[Map[Prepend[#, 0]&, Rest[Map[f, Range[0, nn]!CoefficientList[Series[Exp[y x Exp[x]], {x, 0, nn}], {x, y}]]]], {1}]//Grid (* Geoffrey Critzer, Feb 09 2013 *) t[n_, k_] := Binomial[n, k]*k^(n - k); Prepend[Flatten@Table[t[n, k], {n, 10}, {k, 0, n}], 1] (* Arkadiusz Wesolowski, Mar 23 2013 *)
PROG
(Magma) /* As triangle */ [[Binomial(n, k)*k^(n-k): k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Aug 22 2015 (Sage) # uses[bell_transform from A264428] nat = [k for k in (1..n)]
return bell_transform(n, nat)
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