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

1, 1, 2, 4, 11, 31, 106, 372, 1499, 6211, 28606, 135356, 697357, 3688049, 20935006, 121837276, 753159801, 4767863657, 31807384354, 217048147396, 1551200297291, 11327527814191, 86206555248122, 669666314150164, 5399592811359331, 44398500646885851

Offset: 0

Author

Vladimir Kruchinin, May 24 2011

Keywords

• nonn

Links

• Harvey P. Dale, Table of n, a(n) for n = 0..600

Crossrefs

Column k=4 of A275422.

Programs

Mathematica

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 *)

Maxima

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);

PARI

N=33;  x='x+O('x^N);
egf=exp(x+x^2/2+x^4/4!);
Vec(serlaplace(egf))
/* Joerg Arndt, Sep 15 2012 */

Formula

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

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.

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

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

Extensions

More terms from Harvey P. Dale, Jun 21 2012