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

Author

Karol A. Penson, Mar 04 2003

Keywords

• nonn

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

Crossrefs

Cf. A000110, A001147.

Programs

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 *)

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