A081066 Even order Taylor expansion coefficients at x=0 of exp(exp(x^2/2)-1), odd order coefficients being equal to zero.

1, 1, 6, 75, 1575, 49140, 2110185, 118513395, 8391883500, 728713460475, 75932204473125, 9329869676877750, 1332483237190430325, 218552871240812233125, 40748996386059790578750

Offset

0,3

Comments

From Shai Covo (green355(AT)netvision.net.il), Feb 03 2010: (Start)

Let N be a Poisson random variable with parameter (mean) 1, and Y_1,Y_2,... independent Normal(0,1) (standard normal) random variables, independent of N.

Set S=Sum_{i=1..N} Y_i. Then the moment generating function (MGF) of S is given by exp(exp(x^2/2)-1) (i.e., this is the expectation of exp(xS), x real); hence a(n) is the 2n-th moment of S (the odd moments are equal to zero). More generally, if N above has parameter lambda and Y_i above are Normal(0,sigma^2), then the MGF of S is given by exp(lambda*(exp(sigma^2*x^2/2)-1)) and the 2n-th moment of S is given by (2n-1)!!*sigma^(2n)*Sum_{j=0..n} S2(n,j)*lambda^j, where S2(n,j) are the Stirling numbers of the second kind. (End)

References

S. Covo, The moments of a compound Poisson process with exponential or centered normal jumps, J. Probab. Stat. Sci. 7 (2009), 91-100. [From Shai Covo (green355(AT)netvision.net.il), Feb 03 2010]

Links

Franklin T. Adams-Watters, Jun 18 2008, Table of n, a(n) for n = 0..18

Formula

In Maple notation: a(n)=evalf(subs(x=0, diff((exp(exp(x^2/2)-1), x$2*n)))), n=1, 2...

a(n) = (2*n-1)!!*Bell(n). - Vladeta Jovovic, May 19 2007

E.g.f.: A(x) = exp(-1)*Sum_{n>=0} (1-2*n*x)^(-1/2)/n!. - Vladeta Jovovic, Feb 05 2008

a(n) = A055882(n)*Pochhammer(1/2, n). - Peter Luschny, Nov 07 2011

Maple

A055882 := n-> 2^n*combinat[bell](n);
A081066 := n-> A055882(n)*pochhammer(1/2, n);
seq(A081066(i), i=0..14); # Peter Luschny, Nov 07 2011

Mathematica

Table[(2n-1)!!*BellB[n], {n, 0, 14}] (* Jean-François Alcover, May 23 2016, after Vladeta Jovovic *)

Crossrefs

Cf. A000110, A001147.

Sequence in context: A216136 A360471 A126462 * A185289 A336228 A326011

Adjacent sequences: A081063 A081064 A081065 * A081067 A081068 A081069

Keyword

nonn

Author

Karol A. Penson, Mar 04 2003

Status

approved