A060311 Expansion of e.g.f. exp((exp(x)-1)^2/2).

1, 0, 1, 3, 10, 45, 241, 1428, 9325, 67035, 524926, 4429953, 40010785, 384853560, 3925008361, 42270555603, 478998800290, 5693742545445, 70804642315921, 918928774274028, 12419848913448565, 174467677050577515, 2542777209440690806, 38388037137038323353

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

Column k=2 of A324162.

Cf. A052859, A330047.

Sequence in context: A134018 A355719 A028417 * A184947 A330250 A207652

Adjacent sequences: A060308 A060309 A060310 * A060312 A060313 A060314

Keyword

nonn

Author

Vladeta Jovovic, Mar 27 2001

Status

approved