|
| |
|
|
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, 1, 6, 75, 1575, 49140, 2110185, 118513395, 8391883500, 728713460475, 75932204473125, 9329869676877750, 1332483237190430325, 218552871240812233125, 40748996386059790578750
(list; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 0,3
|
|
|
COMMENTS
| Contribution 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 (FrankTAW(AT)Netscape.net), 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 (vladeta(AT)eunet.rs), May 19 2007
E.g.f.: A(x) = exp(-1)*Sum_{n>=0} (1-2*n*x)^(-1/2)/n!. - Vladeta Jovovic (vladeta(AT)eunet.rs), 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
|
|
|
CROSSREFS
| Cf. A000110, A001147.
Sequence in context: A193784 A162863 A126462 * A185289 A016090 A181343
Adjacent sequences: A081063 A081064 A081065 * A081067 A081068 A081069
|
|
|
KEYWORD
| nonn
|
|
|
AUTHOR
| Karol A. Penson (penson(AT)lptl.jussieu.fr), Mar 04 2003
|
| |
|
|
