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%I #26 Aug 27 2022 18:38:09
%S 1,1,6,75,1575,49140,2110185,118513395,8391883500,728713460475,
%T 75932204473125,9329869676877750,1332483237190430325,
%U 218552871240812233125,40748996386059790578750
%N Even order Taylor expansion coefficients at x=0 of exp(exp(x^2/2)-1), odd order coefficients being equal to zero.
%C From Shai Covo (green355(AT)netvision.net.il), Feb 03 2010: (Start)
%C 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.
%C 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)
%D 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]
%H Franklin T. Adams-Watters, Jun 18 2008, <a href="/A081066/b081066.txt">Table of n, a(n) for n = 0..18</a>
%F In Maple notation: a(n)=evalf(subs(x=0, diff((exp(exp(x^2/2)-1), x$2*n)))), n=1, 2...
%F a(n) = (2*n-1)!!*Bell(n). - _Vladeta Jovovic_, May 19 2007
%F E.g.f.: A(x) = exp(-1)*Sum_{n>=0} (1-2*n*x)^(-1/2)/n!. - _Vladeta Jovovic_, Feb 05 2008
%F a(n) = A055882(n)*Pochhammer(1/2, n). - _Peter Luschny_, Nov 07 2011
%p A055882 := n-> 2^n*combinat[bell](n);
%p A081066 := n-> A055882(n)*pochhammer(1/2,n);
%p seq(A081066(i),i=0..14); # _Peter Luschny_, Nov 07 2011
%t Table[(2n-1)!!*BellB[n], {n, 0, 14}] (* _Jean-François Alcover_, May 23 2016, after _Vladeta Jovovic_ *)
%Y Cf. A000110, A001147.
%K nonn
%O 0,3
%A _Karol A. Penson_, Mar 04 2003
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