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A190452 E.g.f. exp(x+x^2/2+x^4/24).

%I A190452 #21 Jul 27 2016 11:42:06
%S A190452 1,1,2,4,11,31,106,372,1499,6211,28606,135356,697357,3688049,20935006,
%T A190452 121837276,753159801,4767863657,31807384354,217048147396,
%U A190452 1551200297291,11327527814191,86206555248122,669666314150164,5399592811359331,44398500646885851
%N A190452 E.g.f. exp(x+x^2/2+x^4/24).
%H A190452 Harvey P. Dale, <a href="/A190452/b190452.txt">Table of n, a(n) for n = 0..600</a>
%F A190452 E.g.f.: exp(x+x^2/2+x^4/24).
%F A190452 a(n) = n!*sum(k=1..n, sum(j=floor((4*k-n)/3)..floor((4*k-n)/2), binomial(j,n-4*k+3*j)*12^(j-k)*binomial(k,j)*2^(-n+3*k-2*j))/k!), n>0, a(0)=1.
%F A190452 Recurrence: 6*a(n) = 6*a(n-1) + 6*(n-1)*a(n-2) + (n-3)*(n-2)*(n-1)*a(n-4). - _Vaclav Kotesovec_, Oct 09 2013
%F A190452 a(n) ~ 1/2*exp((6*n)^(1/4) + sqrt(6*n)/2 - 3*n/4 - 3/4) * n^(3*n/4) * 6^(-n/4) * (1 + 3^(5/4)/(16*(2*n)^(3/4)) + 7*sqrt(3/2)/(8*sqrt(n)) - 3^(3/4)/(2*(2*n)^(1/4))). - _Vaclav Kotesovec_, Oct 09 2013
%t A190452 With[{nn=30},CoefficientList[Series[Exp[x+x^2/2+x^4/24],{x,0,nn}], x]Range[0,nn]!] (* _Harvey P. Dale_, Jun 21 2012 *)
%o A190452 (Maxima)
%o A190452 a(n):=n!*sum(sum(binomial(j,n-4*k+3*j)*12^(j-k)*binomial(k,j)*2^(-n+3*k-2*j),j,floor((4*k-n)/3),floor((4*k-n)/2))/k!,k,1,n);
%o A190452 (PARI)
%o A190452 N=33;  x='x+O('x^N);
%o A190452 egf=exp(x+x^2/2+x^4/4!);
%o A190452 Vec(serlaplace(egf))
%o A190452 /* _Joerg Arndt_, Sep 15 2012 */
%Y A190452 Column k=4 of A275422.
%K A190452 nonn
%O A190452 0,3
%A A190452 _Vladimir Kruchinin_, May 24 2011
%E A190452 More terms from _Harvey P. Dale_, Jun 21 2012