OFFSET
0,4
COMMENTS
After the first term, this is the Stirling transform of the sequence of moments of the standard normal (or "Gaussian") probability distribution. It is not itself a moment sequence of any probability distribution. - Michael Hardy (hardy(AT)math.umn.edu), May 29 2005
a(n) is the number of simple labeled graphs on n nodes in which each component is a complete bipartite graph. - Geoffrey Critzer, Dec 03 2011
REFERENCES
I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, John Wiley and Sons, N.Y., 1983, Ex. 3.3.5b.
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..518 (first 101 terms from Harry J. Smith)
Paul Barry, Constructing Exponential Riordan Arrays from Their A and Z Sequences, Journal of Integer Sequences, 17 (2014), #14.2.6.
Vaclav Kotesovec, Asymptotic solution of the equations using the Lambert W-function
FORMULA
E.g.f. A(x) = B(exp(x)-1) where B(x)=exp(x^2/2) is e.g.f. of A001147(2n), hence a(n) is the Stirling transform of A001147(2n). - Michael Somos, Jun 01 2005
From Vaclav Kotesovec, Aug 06 2014: (Start)
a(n) ~ exp(1/2*(exp(r)-1)^2 - n) * n^(n+1/2) / (r^n * sqrt(exp(r)*r*(-1-r+exp(r)*(1+2*r)))), where r is the root of the equation exp(r)*(exp(r) - 1)*r = n.
(a(n)/n!)^(1/n) ~ 2*exp(1/LambertW(2*n)) / LambertW(2*n).
(End)
a(n) = Sum_{k=0..floor(n/2)} (2*k)! * Stirling2(n,2*k)/(2^k * k!). - Seiichi Manyama, May 07 2022
MAPLE
a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)
*binomial(n-1, j-1)*Stirling2(j, 2), j=2..n))
end:
seq(a(n), n=0..25); # Alois P. Heinz, Sep 02 2019
MATHEMATICA
a = Exp[x] - 1; Range[0, 20]! CoefficientList[Series[Exp[a^2/2], {x, 0, 20}], x] (* Geoffrey Critzer, Dec 03 2011 *)
PROG
(PARI) a(n)=if(n<0, 0, n!*polcoeff( exp((exp(x+x*O(x^n))-1)^2/2), n)) /* Michael Somos, Jun 01 2005 */
(PARI) { for (n=0, 100, write("b060311.txt", n, " ", n!*polcoeff(exp((exp(x + x*O(x^n)) - 1)^2/2), n)); ) } \\ Harry J. Smith, Jul 03 2009
(PARI) a(n) = sum(k=0, n\2, (2*k)!*stirling(n, 2*k, 2)/(2^k*k!)); \\ Seiichi Manyama, May 07 2022
CROSSREFS
KEYWORD
nonn
AUTHOR
Vladeta Jovovic, Mar 27 2001
STATUS
approved