A190452 - OEIS

%I #21 Jul 27 2016 11:42:06

%S 1,1,2,4,11,31,106,372,1499,6211,28606,135356,697357,3688049,20935006,

%T 121837276,753159801,4767863657,31807384354,217048147396,

%U 1551200297291,11327527814191,86206555248122,669666314150164,5399592811359331,44398500646885851

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

%H Harvey P. Dale, <a href="/A190452/b190452.txt">Table of n, a(n) for n = 0..600</a>

%F E.g.f.: exp(x+x^2/2+x^4/24).

%F 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 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 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 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 (Maxima)

%o 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 (PARI)

%o N=33;  x='x+O('x^N);

%o egf=exp(x+x^2/2+x^4/4!);

%o Vec(serlaplace(egf))

%o /* _Joerg Arndt_, Sep 15 2012 */

%Y Column k=4 of A275422.

%K nonn

%O 0,3

%A _Vladimir Kruchinin_, May 24 2011

%E More terms from _Harvey P. Dale_, Jun 21 2012