OFFSET
0,3
COMMENTS
Number of forests with n nodes and height at most 1.
Equivalently, number of idempotent mappings f from a set of n elements into itself (i.e., satisfying f o f = f). - Robert FERREOL, Oct 11 2007
In other words, a(n) = number of idempotents in the full semigroup of maps from [1..n] to itself. [Tainiter]
a(n) is the number of ways to select a set partition of {1,2,...,n} and then designate one element in each block (cell) of the partition.
Let set B have cardinality n. Then a(n) is the number of functions f:D->C over all partitions {D,C} of B. See the example in the Example Section below. We note that f:empty set->B is designated as the null function, whereas f:B->empty set is undefined unless B itself is empty. - Dennis P. Walsh, Dec 05 2013
In physics, a(n) would be interpreted as the number of projection operators P on S_n, i.e., ones satisfying P^2 = P. Example: a particle with a half-integer spin s has a spin space with 2s+1 base states which admits a(2s+1) linear projection operators (including the identity). These are important because they satisfy the operator identity exp(zU) = 1+(exp(z)-1)*U, valid for any complex z. - Stanislav Sykora, Nov 03 2016
REFERENCES
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 91.
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. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.32(d).
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..541 (first 101 terms from T. D. Noe)
Alexander Burstein and Louis W. Shapiro, Pseudo-involutions in the Riordan group, arXiv:2112.11595 [math.CO], 2021.
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 131.
Xing Gao and William F. Keigher, Interlacing of Hurwitz series, Communications in Algebra, 45:5 (2017), 2163-2185, DOI: 10.1080/00927872.2016.1226885. See Ex. 2.13.
B. Harris and L. Schoenfeld, The number of idempotent elements in symmetric semigroups, J. Combin. Theory, 3 (1967), 122-135.
Bernard Harris and Lowell Schoenfeld, Asymptotic expansions for the coefficients of analytic functions, Illinois Journal of Mathematics, Volume 12, Issue 2 (1968), 264-277.
G. Helms, Pascalmatrix tetrated [From Gottfried Helms, Feb 04 2009]
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 117
Vaclav Kotesovec, Graph - the asymptotic ratio (1000 terms)
Nate Kube and Frank Ruskey, Sequences That Satisfy a(n-a(n))=0, Journal of Integer Sequences, Vol. 8 (2005), Article 05.5.5.
J. Riordan, Forests of labeled trees, J. Combin. Theory, 5 (1968), 90-103.
J. Riordan, Letter to N. J. A. Sloane, Oct. 1970
John Riordan and N. J. A. Sloane, Correspondence, 1974
Emre Sen, Exceptional Sequences and Idempotent Functions, arXiv:1909.05887 [math.RT], 2019.
M. Tainiter, A characterization of idempotents in semigroups, J. Combinat. Theory, 5 (1968), 370-373.
Haoliang Wang and Robert Simon, The Analysis of Synchronous All-to-All Communication Protocols for Wireless Systems, Q2SWinet'18: Proceedings of the 14th ACM International Symposium on QoS and Security for Wireless and Mobile Networks (2018), 39-48.
FORMULA
G.f.: Sum_{k>=0} x^k/(1-k*x)^(k+1). - Vladeta Jovovic, Oct 25 2003
a(n) = Sum_{k=0..n} C(n,k)*(n-k)^k. - Paul D. Hanna, Jun 26 2009
G.f.: G(0) where G(k) = 1 - x*(-1+2*k*x)^(2*k+1)/((x-1+2*k*x)^(2*k+2) - x*(x-1+2*k*x)^(4*k+4)/(x*(x-1+2*k*x)^(2*k+2) - (2*x-1+2*k*x)^(2*k+3)/G(k+1))) (continued fraction). - Sergei N. Gladkovskii, Jan 26 2013
E.g.f.: 1 + x/(1+x)*(G(0) - 1) where G(k) = 1 + exp(x)/(k+1)/(1-x/(x+(1)/G(k+1))) (continued fraction). - Sergei N. Gladkovskii, Feb 04 2013
Recurrence: a(0)=1, a(n) = Sum_{k=1..n} binomial(n-1,k-1)*k*a(n-k). - James East, Mar 30 2014
Asymptotics (Harris and Schoenfeld, 1968): a(n) ~ sqrt((r+1)/(2*Pi*(n+1)*(r^2+3*r+1))) * n! * exp((n+1)/(r+1)) / r^n, where r is the root of the equation r*(r+1)*exp(r) = n+1. - Vaclav Kotesovec, Jul 13 2014
a(n) = Sum_{k=0..n} A005727(k)*Stirling2(n, k). - Mélika Tebni, Jun 12 2022
More precise asymptotics: a(n) ~ n^(n + 1/2) / (sqrt(1 + 3*r + r^2) * exp(n - r*exp(r) + r/2) * r^(n + 1/2)), where r = 2*w - 1/(2*w) + 5/(8*w^2) - 19/(24*w^3) + 209/(192*w^4) - 763/(480*w^5) + 4657/(1920*w^6) - 6855/(1792*w^7) + 199613/(32256*w^8) + ... and w = LambertW(sqrt(n)/2). - Vaclav Kotesovec, Feb 20 2023
EXAMPLE
a(3)=10 since, for B={1,2,3}, we have 10 functions: 1 function of the type f:empty set->B; 6 functions of the type f:{x}->B\{x}; and 3 functions of the type f:{x,y}->B\{x,y}. - Dennis P. Walsh, Dec 05 2013
MAPLE
A000248 := proc(n) local k; add(k^(n-k)*binomial(n, k), k=0..n); end; # Robert FERREOL, Oct 11 2007
a:= proc(n) option remember; if n=0 then 1 else add(binomial(n-1, j) *(j+1) *a(n-1-j), j=0..n-1) fi end: seq(a(n), n=0..20); # Zerinvary Lajos, Mar 28 2009
MATHEMATICA
CoefficientList[Series[Exp[x Exp[x]], {x, 0, 20}], x]*Table[n!, {n, 0, 20}]
a[0] = 1; a[1] = 1; a[n_] := a[n] = a[n - 1] + Sum[(Binomial[n - 1, j] + (n - 1) Binomial[n - 2, j]) a[j], {j, 0, n - 2}]; Table[a[n], {n, 0, 20}] (* David Callan, Oct 04 2013 *)
Flatten[{1, Table[Sum[Binomial[n, k]*(n-k)^k, {k, 0, n}], {n, 1, 20}]}] (* Vaclav Kotesovec, Jul 13 2014 *)
Table[Sum[BellY[n, k, Range[n]], {k, 0, n}], {n, 0, 20}] (* Vladimir Reshetnikov, Nov 09 2016 *)
PROG
(PARI) a(n)=sum(k=0, n, binomial(n, k)*(n-k)^k); \\ Paul D. Hanna, Jun 26 2009
(PARI) x='x+O('x^66); Vec(serlaplace(exp(x*exp(x)))) \\ Joerg Arndt, Oct 06 2013
(Sage) # uses[bell_matrix from A264428]
B = bell_matrix(lambda k: k+1, 20)
print([sum(B.row(n)) for n in range(20)]) # Peter Luschny, Sep 03 2019
(Magma) m:=25; R<x>:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(Exp(x*Exp(x)))); [Factorial(n-1)*b[n]: n in [1..m]]; // Vincenzo Librandi, Feb 01 2020
CROSSREFS
KEYWORD
easy,nonn,nice
AUTHOR
EXTENSIONS
In view of the multiple appearances of this sequence, I replaced the definition with the simple exponential generating function. - N. J. A. Sloane, Apr 16 2018
STATUS
approved