| Displaying 1-10 of 10 results found.
page
1
1, 1, 5, 33, 273, 2721, 31701, 421905, 6302913, 104270913, 1889862021, 37204038081, 789866524305, 17977594555233, 436435929785493, 11251798888929201, 306889765901872641, 8825681949708120705, 266828094135981378693, 8458295877281844310113
FORMULA
E.g.f. exp(p(x)) with p(x) := x*(3-3*x+x^2)/(3*(1-x)^3) (E.g.f. first column of A049352). a(n) ~ n^(n-1/8)/2 * exp(-1/4 + 5*n^(1/4)/24 + sqrt(n)/2 + 4*n^(3/4)/3 - n). - Vaclav Kotesovec, Oct 23 2017 E.g.f.: Sum_{n>=0} ( Integral 1/(1-x)^4 dx )^n / n!, where the constant of integration is taken to be zero. - Paul D. Hanna, Apr 27 2019
MAPLE
a:= proc(n) option remember; `if`(n=0, 1, add(
binomial(n-1, j-1)*(j+2)!/6*a(n-j), j=1..n))
end:
MATHEMATICA
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n-1, j-1]*(j+2)!/6*a[n-j], {j, 1, n}];
Partition number array, called M31(4), related to A049352(n,m)= |S1(4;n,m)| (generalized Stirling triangle).
+20
4
1, 4, 1, 20, 12, 1, 120, 80, 48, 24, 1, 840, 600, 800, 200, 240, 40, 1, 6720, 5040, 7200, 4000, 1800, 4800, 960, 400, 720, 60, 1, 60480, 47040, 70560, 84000, 17640, 50400, 28000, 33600, 4200, 16800, 6720, 700, 1680, 84, 1, 604800, 483840, 752640, 940800, 504000, 188160
COMMENTS
Each partition of n, ordered as in Abramowitz-Stegun (A-St order; for the reference see A134278), is mapped to a nonnegative integer a(n,k) =: M31(4;n,k) with the k-th partition of n in A-St order. The sequence of row lengths is A000041 (partition numbers) [1, 2, 3, 5, 7, 11, 15, 22, 30, 42,...]. Fourth member (K=4) in the family M31(K) of partition number arrays.
If M31(4;n,k) is summed over those k with fixed number of parts m one obtains the unsigned triangle |S1(4)|:= A049352.
FORMULA
a(n,k)=(n!/product(e(n,k,j)!*j!^(e(n,k,j),j=1..n))*product(|S1(4;j,1)|^e(n,k,j),j =1..n) = M3(n,k)*product(|S1(4;j,1)|^e(n,k,j),j=1..n) with |S1(4;n,1)|= A001715(n+2) = (n+2)!/3!, n>=1 and the exponent e(n,k,j) of j in the k-th partition of n in the A-St ordering of the partitions of n. M3(n,k)=A036040.
EXAMPLE
[1];[4,1];[20,12,1];[120,80,48,24,1];[840,600,800,200,240,40,1];...
a(4,3)= 48 = 3*|S1(4;2,1)|^2. The relevant partition of 4 is (2^2).
Alternating row sums of triangle A049352 (S1p(4)).
+20
1
1, 3, 9, 15, -159, -3021, -36903, -381249, -3212415, -12995901, 315857961, 12457515663, 304888863969, 6280156107315, 113710631625081, 1717208752084479, 15528594345353217, -265033870991715069, -22048996644203788215
FORMULA
a(n)=sum( A049352(n,m)*(-1)^(m-1),m=1..n), n>=1. E.g.f.: 1-exp(-x*(3-3*x+x^2)/(3*(1-x)^3)). Cf. e.g.f. first column of A049352.
a(n) = n!/6.
(Formerly M3566 N1445)
+10
51
1, 4, 20, 120, 840, 6720, 60480, 604800, 6652800, 79833600, 1037836800, 14529715200, 217945728000, 3487131648000, 59281238016000, 1067062284288000, 20274183401472000, 405483668029440000, 8515157028618240000, 187333454629601280000, 4308669456480829440000
COMMENTS
The numbers (4, 20, 120, 840, 6720, ...) arise from the divisor values in the general formula a(n) = n*(n+1)*(n+2)*(n+3)* ... *(n+k)*(n*(n+k) + (k-1)*k/6)/((k+3)!/6) (which covers the following sequences: A000578, A000537, A024166, A101094, A101097, A101102). - Alexander R. Povolotsky, May 17 2008 a(n) is also the number of decreasing 3-cycles in the decomposition of permutations as product of disjoint cycles, a(3)=1, a(4)=4, a(5)=20. - Wenjin Woan, Dec 21 2008 a(n) is the number of n-permutations having 1, 2, and 3 in three distinct cycles. - Geoffrey Critzer, Apr 26 2009 The asymptotic expansion of the higher order exponential integral E(x,m=1,n=4) ~ exp(-x)/x*(1 - 4/x + 20/x^2 - 120/x^3 + 840/x^4 - 6720/x^5 + 60480/x^6 - 604800/x^7 + ...) leads to the sequence given above. See A163931 and A130534 for more information. (End)
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
a(n) = A049352(n-2, 1) (first column of triangle). E.g.f. if offset 0: 1/(1-x)^4.
G.f.: G(0)/2, where G(k) = 1 + 1/(1 - x/(x + 1/(k+4)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 01 2013 G.f.: W(0), where W(k) = 1 - x*(k+4)/( x*(k+4) - 1/(1 - x*(k+1)/( x*(k+1) - 1/W(k+1) ))); (continued fraction). - Sergei N. Gladkovskii, Aug 26 2013 The o.g.f. A(x) satisfies the Riccati equation x^2*A'(x) + (4*x - 1)*A(x) + 1 = 0.
G.f. as an S-fraction: A(x) = 1/(1 - 4*x/(1 - x/(1 - 5*x/(1 - 2*x/(1 - 6*x/(1 - 3*x/(1 - ... - (n + 3)*x/(1 - n*x/(1 - ... ))))))))) (apply Stokes, 1982).
A(x) = 1/(1 - 3*x - x/(1 - 4*x/(1 - 2*x/(1 - 5*x/(1 - 3*x/(1 - 6*x/(1 - ... - n*x/(1 - (n+3)*x/(1 - ... ))))))))). (End)
H(x) = (1 - (1 + x)^(-3)) / 3 = x - 4 x^2/2! + 20 x^3/3! - ... is an e.g.f. of the signed sequence (n!/4!), which is the compositional inverse of G(x) = (1 - 3*x)^(-1/3) - 1, an e.g.f. for A007559. Cf. A094638, A001710 (for n!/2!), and A001720 (for n!/4!). Cf. columns of A094587, A173333, and A213936 and rows of A138533.- Tom Copeland, Dec 27 2019 Sum_{n>=3} 1/a(n) = 6*e - 15.
Sum_{n>=3} (-1)^(n+1)/a(n) = 3 - 6/e. (End)
MAPLE
f := proc(n) n!/6; end;
BB:= [S, {S = Prod(Z, Z, C), C = Union(B, Z, Z), B = Prod(Z, C)}, labelled]: seq(combstruct[count](BB, size=n)/12, n=3..20); # Zerinvary Lajos, Jun 19 2008 G(x):=1/(1-x)^4: f[0]:=G(x): for n from 1 to 18 do f[n]:=diff(f[n-1], x) od: x:=0: seq(f[n], n=0..16); # Zerinvary Lajos, Apr 01 2009
A partition product of Stirling_1 type [parameter k = -4] with biggest-part statistic (triangle read by rows).
+10
11
1, 1, 4, 1, 12, 20, 1, 72, 80, 120, 1, 280, 1000, 600, 840, 1, 1740, 9200, 9000, 5040, 6720, 1, 8484, 79100, 138600, 88200, 47040, 60480, 1, 57232, 874720, 1789200, 1552320, 940800, 483840, 604800, 1, 328752, 9532880
COMMENTS
Partition product of prod_{j=0..n-2}(k-n+j+2) and n! at k = -4,
summed over parts with equal biggest part (see the Luschny link).
Underlying partition triangle is A144354. Same partition product with length statistic is A049352.
FORMULA
T(n,0) = [n = 0] (Iverson notation) and for n > 0 and 1 <= m <= n
T(n,m) = Sum_{a} M(a)|f^a| where a = a_1,..,a_n such that
1*a_1+2*a_2+...+n*a_n = n and max{a_i} = m, M(a) = n!/(a_1!*..*a_n!),
f^a = (f_1/1!)^a_1*..*(f_n/n!)^a_n and f_n = product_{j=0..n-2}(j-n-2).
A triangle of numbers related to triangle A030526.
+10
10
1, 5, 1, 30, 15, 1, 210, 195, 30, 1, 1680, 2550, 675, 50, 1, 15120, 34830, 14025, 1725, 75, 1, 151200, 502740, 287280, 51975, 3675, 105, 1, 1663200, 7692300, 5961060, 1482705, 151200, 6930, 140, 1, 19958400, 124740000, 126913500, 41545980
COMMENTS
a(n,1)= A001720(n+3). a(n,m)=: S1p(5; n,m), a member of a sequence of lower triangular Jabotinsky matrices with nonnegative entries, including S1p(1; n,m)= A008275 (unsigned Stirling first kind), S1p(2; n,m)= A008297(n,m) (unsigned Lah numbers), S1p(3; n,m)= A046089(n,m), S1p(4; n,m)= A049352(n,m). Signed lower triangular matrix (-1)^(n-m)*a(n,m) is inverse to matrix A049029(n,m) := S2(5; n,m). The monic row polynomials E(n,x) := sum(a(n,m)*x^m,m=1..n), E(0,x) := 1 are exponential convolution polynomials (see A039692 for the definition and a Knuth reference). a(n,m) enumerates unordered increasing n-vertex m-forests composed of m unary trees (out-degree r from {0,1}) whose vertices of depth (distance from the root) j>=1 come in j+4 colors. The k roots (j=0) each come in one (or no) color. - Wolfdieter Lang, Oct 12 2007
FORMULA
a(n, m) = n!* A030526(n, m)/(m!*4^(n-m)); a(n, m) = (4*m+n-1)*a(n-1, m) + a(n-1, m-1), n >= m >= 1; a(n, m)=0, n<m; a(n, 0) := 0; a(1, 1)=1. E.g.f. for m-th column: ((x*(2-x)*(2-2*x+x^2)/(4*(1-x)^4))^m)/m!. a(n,k) = (n!*sum(j=1..k, (-1)^(k-j)*binomial(k,j)*binomial(n+4*j-1,4*j-1)))/(4^k*k!). - Vladimir Kruchinin, Apr 01 2011
EXAMPLE
Triangle begins:
{1};
{5,1};
{30,15,1}; E.g., row polynomial E(3,x)=30*x+15*x^2+x^3.
{210,195,30,1};
...
a(4,2)= 195 =4*(5*6)+3*(5*5) from the two types of unordered 2-forests of unary increasing trees associated with the two m=2 parts partitions (1,3) and (2^2) of n=4. The first type has 4 increasing labelings, each coming in (1)*(1*5*6)=30 colored versions, e.g., ((1c1),(2c1,3c5,4c6)) with lcp for vertex label l and color p. Here the vertex labeled 3 has depth j=1, hence 5 colors, c1..c5, can be chosen and the vertex labeled 4 with j=2 can come in 6 colors, e.g., c1..c6. Therefore there are 4*((1)*(1*5*6))=120 forests of this (1,3) type. Similarly the (2,2) type yields 3*((1*5)*(1*5))=75 such forests, e.g., ((1c1,3c4)(2c1,4c5)) or ((1c1,3c5)(2c1,4c2)), etc. - Wolfdieter Lang, Oct 12 2007
MAPLE
# The function BellMatrix is defined in A264428. # Adds (1, 0, 0, 0, ..) as column 0.
MATHEMATICA
a[n_, m_] /; n >= m >= 1 := a[n, m] = (4m + n - 1)*a[n-1, m] + a[n-1, m-1]; a[n_, m_] /; n < m = 0; a[_, 0] = 0; a[1, 1] = 1; Flatten[Table[a[n, m], {n, 1, 9}, {m, 1, n}]] (* Jean-François Alcover, Jul 22 2011 *) BellMatrix[f_Function, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len-1}, {k, 0, len-1}]];
rows = 10;
M = BellMatrix[(#+4)!/24&, rows];
PROG
(Maxima) a(n, k):=(n!*sum((-1)^(k-j)*binomial(k, j)*binomial(n+4*j-1, 4*j-1), j, 1, k))/(4^k*k!); /* Vladimir Kruchinin, Apr 01 2011 */
CROSSREFS
Cf. A134139 (alternating row sums).
A triangle of numbers related to triangle A030527.
+10
9
1, 6, 1, 42, 18, 1, 336, 276, 36, 1, 3024, 4200, 960, 60, 1, 30240, 66024, 23400, 2460, 90, 1, 332640, 1086624, 557424, 87360, 5250, 126, 1, 3991680, 18805248, 13349952, 2916144, 255360, 9912, 168, 1, 51891840, 342486144, 325854144, 95001984
COMMENTS
a(n,1) = A001725(n+4). a(n,m)=: S1p(6; n,m), a member of a sequence of lower triangular Jabotinsky matrices with nonnegative entries, including S1p(1; n,m) = A008275 (unsigned Stirling first kind), S1p(2; n,m) = A008297(n,m) (unsigned Lah numbers). S1p(3; n,m) = A046089(n,m), S1p(4; n,m) = A049352, S1p(5; n,m) = A049353(n,m). Signed lower triangular matrix (-1)^(n-m)*a(n,m) is inverse to matrix A049385(n,m) =: S2(6; n,m). The monic row polynomials E(n,x) := Sum_{m=1..n} (a(n,m)*x^m), E(0,x) := 1 are exponential convolution polynomials (see A039692 for the definition and a Knuth reference). a(n,m) enumerates unordered increasing n-vertex m-forests composed of m unary trees (out-degree r from {0,1}) whose vertices of depth (distance from the root) j >= 1 come in j+5 colors. The k roots (j=0) each come in one (or no) color. - Wolfdieter Lang, Oct 12 2007
FORMULA
a(n, m) = n!* A030527(n, m)/(m!*5^(n-m)); a(n, m) = (5*m+n-1)*a(n-1, m) + a(n-1, m-1), n >= m >= 1; a(n, m)=0, n < m; a(n, 0) := 0; a(1, 1)=1. E.g.f. for m-th column: ((x*(5 - 10*x + 10*x^2 - 5*x^3 + x^4)/(5*(1-x)^5))^m)/m!. a(n,k) = n!* Sum_{j=1..k} (-1)^(k-j)*binomial(k,j)*binomial(n+5*j-1,5*j-1) /(5^k*k!). - Vladimir Kruchinin, Apr 01 2011
EXAMPLE
Triangle begins
1;
6, 1;
42, 18, 1;
336, 276, 36, 1;
3024, 4200, 960, 60, 1;
30240, 66024, 23400, 2460, 90, 1;
332640, 1086624, 557424, 87360, 5250, 126, 1;
E.g., row polynomial E(3,x) = 42*x + 18*x^2 + x^3.
a(4,2) = 276 = 4*(6*7) + 3*(6*6) from the two types of unordered 2-forests of unary increasing trees associated with the two m=2 parts partitions (1,3) and (2^2) of n=4. The first type has 4 increasing labelings, each coming in (1)*(1*6*7)=42 colored versions, e.g., ((1c1),(2c1,3c6,4c3)) with lcp for vertex label l and color p. Here the vertex labeled 3 has depth j=1, hence 6 colors, c1..c6, can be chosen and the vertex labeled 4 with j=2 can come in 7 colors, e.g., c1..c7. Therefore there are 4*((1)*(1*6*7))=168 forests of this (1,3) type. Similarly the (2,2) type yields 3*((1*6)*(1*6))=108 such forests, e.g., ((1c1,3c4)(2c1,4c6)) or ((1c1,3c5)(2c1,4c2)), etc. - Wolfdieter Lang, Oct 12 2007
MAPLE
# The function BellMatrix is defined in A264428. # Adds (1, 0, 0, 0, ..) as column 0.
MATHEMATICA
a[n_, k_] = n!*Sum[(-1)^(k-j)*Binomial[k, j]*Binomial[n + 5j - 1, 5j - 1]/(5^k*k!), {j, 1, k}] ;
Flatten[Table[a[n, k], {n, 1, 9}, {k, 1, n}] ][[1 ;; 40]]
BellMatrix[f_Function, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len-1}, {k, 0, len-1}]];
rows = 10;
M = BellMatrix[(#+5)!/120&, rows];
PROG
(Maxima)
a(n, k)=(n!*sum((-1)^(k-j)*binomial(k, j)*binomial(n+5*j-1, 5*j-1), j, 1, k))/(5^k*k!); /* Vladimir Kruchinin, Apr 01 2011 */ (PARI)
a(n, k)=(n!*sum(j=1, k, (-1)^(k-j)*binomial(k, j)*binomial(n+5*j-1, 5*j-1)))/(5^k*k!);
for(n=1, 12, for(k=1, n, print1(a(n, k), ", ")); print()); /* print triangle */ /* Joerg Arndt, Apr 01 2011 */ (GAP) Flat(List([1..10], n->Factorial(n)*List([1..n], k->Sum([1..k], j->(-1)^(k-j)*Binomial(k, j)*Binomial(n+5*j-1, 5*j-1)/(5^k*Factorial(k)))))); # Muniru A Asiru, Jun 23 2018
Lower triangular array called S1hat(4) related to partition number array A144885.
+10
5
1, 4, 1, 20, 4, 1, 120, 36, 4, 1, 840, 200, 36, 4, 1, 6720, 1720, 264, 36, 4, 1, 60480, 12480, 2040, 264, 36, 4, 1, 604800, 118560, 16000, 2296, 264, 36, 4, 1, 6652800, 1081920, 149600, 17280, 2296, 264, 36, 4, 1, 79833600, 11793600, 1362240, 163680, 18304, 2296, 264
COMMENTS
If in the partition array M31hat(4):= A144885 entries with the same parts number m are summed one obtains this triangle of numbers S1hat(4). In the same way the signless Stirling1 triangle | A008275| is obtained from the partition array M_2 = A036039.
FORMULA
a(n,m)=sum(product(|S1(4;j,1)|^e(n,m,q,j),j=1..n),q=1..p(n,m)) if n>=m>=1, else 0. Here p(n,m)= A008284(n,m), the number of m parts partitions of n and e(n,m,q,j) is the exponent of j in the q-th m part partition of n. |S1(4,n,1)|= A049352(n,1) = A001715(n+2) = (n+2)!/3!.
EXAMPLE
[1];[4,1];[20,4,1];[120,36,4,1];[840,200,36,4,1];...
Generalized unsigned Stirling1 triangle, S1p(7).
+10
4
1, 7, 1, 56, 21, 1, 504, 371, 42, 1, 5040, 6440, 1295, 70, 1, 55440, 114520, 36225, 3325, 105, 1, 665280, 2116800, 983920, 135975, 7105, 147, 1, 8648640, 40884480, 26714800, 5199145, 398860, 13426, 196, 1, 121080960, 826338240, 735469280
COMMENTS
Signed lower triangular matrix (-1)^(n-m)*a(n,m) is inverse to matrix A092082(n, m) =: S2(7; n,m). The monic row polynomials E(n,x) := sum(a(n,m)*x^m, m=1..n), E(0,x) := 1 are exponential convolution polynomials (see A039692 for the definition and a Knuth reference). a(n,m) enumerates unordered increasing n-vertex m-forests composed of m unary trees (out-degree r from {0,1}) whose vertices of depth (distance from the root) j>=1 come in j+6 colors. The k roots (j=0) each come in one (or no) color. - Wolfdieter Lang, Oct 05 2007 A triangle of numbers related to triangle A132166. a(n,1)= A001730(n,5), n>=1. a(n,m)=: S1p(7; n,m), a member of a sequence of lower triangular Jabotinsky matrices with nonnegative entries, including S1p(1; n, m)= A008275 (unsigned Stirling first kind), S1p(2; n,m)= A008297(n, m) (unsigned Lah numbers). S1p(3; n,m)= A046089(n,m), S1p(4; n,m)= A049352, S1p(5; n,m)= A049353(n,m), S1p(6; n,m)= A049374(n, m). The Bell transform of factorial(n+6)/factorial(6). For the definition of the Bell transform see A264428. - Peter Luschny, Jan 18 2016
FORMULA
a(n, m) = n!* A132166(n, m)/(m!*6^(n-m)); a(n, m) = (6*m+n-1)*a(n-1, m) + a(n-1, m-1), n >= m >= 1; a(n, m)=0, n<m; a(n, 0) := 0; a(1, 1)=1. E.g.f. for m-th column: ((x*(6-15*x+20*x^2-15*x^3+6*x^4-x^5)/(6*(1-x)^6))^m)/m!.
EXAMPLE
{1}; {7,1}; {56,21,1}; {504,371,42,1}; ... E.g. Row polynomial E(3,x)=56*x+21*x^2+x^3.
a(4,2)= 371 = 4*(7*8)+3*(7*7) from the two types of unordered 2-forests of unary increasing trees associated with the two m=2 parts partitions (1,3) and (2^2) of n=4. The first type has 4 increasing labelings, each coming in (1)*(1*7*8)=56 colored versions, e.g., ((1c1),(2c1,3c7,4c5)) with lcp for vertex label l and color p. Here the vertex labeled 3 has depth j=1, hence 7 colors, c1..c7, can be chosen and the vertex labeled 4 with j=2 can come in 8 colors, e.g., c1..c8. Therefore there are 4*((1)*(1*7*8))=224 forests of this (1,3) type. Similarly the (2,2) type yields 3*((1*7)*(1*7))=147 such forests, e.g. ((1c1,3c4)(2c1,4c7)) or ((1c1,3c6)(2c1,4c2)), etc. - Wolfdieter Lang, Oct 05 2007
MAPLE
# The function BellMatrix is defined in A264428. # Adds (1, 0, 0, 0, ..) as column 0.
MATHEMATICA
a[n_, m_] /; n >= m >= 1 := a[n, m] = (6*m + n - 1)*a[n-1, m] + a[n-1, m-1]; a[n_, m_] /; n < m = 0; a[_, 0] = 0; a[1, 1] = 1; Flatten[Table[a[n, m], {n, 1, 9}, {m, 1, n}]][[1 ;; 39]] (* Jean-François Alcover, Jun 01 2011, after formula *) BellMatrix[f_Function, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
rows = 12;
M = BellMatrix[(# + 6)!/6! &, rows];
PROG
(Sage) # uses[bell_matrix from A264428] # Adds a column 1, 0, 0, 0, ... at the left side of the triangle.
bell_matrix(lambda n: factorial(n+6)/factorial(6), 10) # Peter Luschny, Jan 18 2016
Partition number array, called M31hat(4).
+10
4
1, 4, 1, 20, 4, 1, 120, 20, 16, 4, 1, 840, 120, 80, 20, 16, 4, 1, 6720, 840, 480, 400, 120, 80, 64, 20, 16, 4, 1, 60480, 6720, 3360, 2400, 840, 480, 400, 320, 120, 80, 64, 20, 16, 4, 1, 604800, 60480, 26880, 16800, 14400, 6720, 3360, 2400, 1920, 1600, 840, 480, 400, 320
COMMENTS
Each partition of n, ordered as in Abramowitz-Stegun (A-St order; for the reference see A134278), is mapped to a nonnegative integer a(n,k) =: M31hat(4;n,k) with the k-th partition of n in A-St order. The sequence of row lengths is A000041 (partition numbers) [1, 2, 3, 5, 7, 11, 15, 22, 30, 42,...]. Fourth member (K=4) in the family M31hat(K) of partition number arrays.
If M31hat(4;n,k) is summed over those k with fixed number of parts m one obtains the unsigned triangle S1hat(4):= A144886.
FORMULA
a(n,k) = product(|S1(4;j,1)|^e(n,k,j),j=1..n) with |S1(4;n,1)| = A049352(n,1) = A001715(n+2) = [1,4,20,120,840,6720,...] = (n+2)!/3!, n>=1 and the exponent e(n,k,j) of j in the k-th partition of n in the A-St ordering of the partitions of n.
EXAMPLE
[1];[4,1];[20,4,1];[120,20,16,4,1];[840,120,80,20,16,4,1];...
a(4,3)= 16 = |S1(4;2,1)|^2. The relevant partition of 4 is (2^2).
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