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%I A060311 #47 May 08 2022 08:24:47
%S A060311 1,0,1,3,10,45,241,1428,9325,67035,524926,4429953,40010785,384853560,
%T A060311 3925008361,42270555603,478998800290,5693742545445,70804642315921,
%U A060311 918928774274028,12419848913448565,174467677050577515,2542777209440690806,38388037137038323353
%N A060311 Expansion of e.g.f. exp((exp(x)-1)^2/2).
%C A060311 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
%C A060311 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
%D A060311 I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, John Wiley and Sons, N.Y., 1983, Ex. 3.3.5b.
%H A060311 Alois P. Heinz, <a href="/A060311/b060311.txt">Table of n, a(n) for n = 0..518</a> (first 101 terms from Harry J. Smith)
%H A060311 Paul Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Barry2/barry281.html">Constructing Exponential Riordan Arrays from Their A and Z Sequences</a>, Journal of Integer Sequences, 17 (2014), #14.2.6.
%H A060311 Vaclav Kotesovec, <a href="http://oeis.org/A216688/a216688.pdf">Asymptotic solution of the equations using the Lambert W-function</a>
%F A060311 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
%F A060311 From _Vaclav Kotesovec_, Aug 06 2014: (Start)
%F A060311 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.
%F A060311 (a(n)/n!)^(1/n) ~ 2*exp(1/LambertW(2*n)) / LambertW(2*n).
%F A060311 (End)
%F A060311 a(n) = Sum_{k=0..floor(n/2)} (2*k)! * Stirling2(n,2*k)/(2^k * k!). - _Seiichi Manyama_, May 07 2022
%p A060311 a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)
%p A060311       *binomial(n-1, j-1)*Stirling2(j, 2), j=2..n))
%p A060311     end:
%p A060311 seq(a(n), n=0..25);  # _Alois P. Heinz_, Sep 02 2019
%t A060311 a = Exp[x] - 1; Range[0, 20]! CoefficientList[Series[Exp[a^2/2], {x, 0, 20}], x] (* _Geoffrey Critzer_, Dec 03 2011 *)
%o A060311 (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 */
%o A060311 (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
%o A060311 (PARI) a(n) = sum(k=0, n\2, (2*k)!*stirling(n, 2*k, 2)/(2^k*k!)); \\ _Seiichi Manyama_, May 07 2022
%Y A060311 Column k=2 of A324162.
%Y A060311 Cf. A052859, A330047.
%K A060311 nonn
%O A060311 0,4
%A A060311 _Vladeta Jovovic_, Mar 27 2001

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