A112242 E.g.f. exp( x*(1+x)/(1-x) ).

1, 1, 5, 25, 169, 1361, 12781, 136585, 1633745, 21594529, 312239701, 4898379641, 82810239865, 1500015354865, 28970936174909, 594083320767721, 12886811207794081, 294742414455540545, 7087332775240107685, 178707496551303048409, 4714241296084031285321, 129830157857411005318801

Offset: 0

Author

Paul Barry, Aug 29 2005

Keywords

• easy
• nonn

Comments

In general, e.g.f. exp(x*(1+a*x)/(1-b*x)) has general term sum{i=0..n, sum{j=0..n, a^j*b^(n-i-j)*C(i,j)*C(n-j-1,n-i-j)*n!/i!}}.

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..441

Programs

Mathematica

Range[0, 18]!*CoefficientList[ Series[ Exp[x(1+x)/(1-x)], {x, 0, 18}], x] (* Zerinvary Lajos, Mar 23 2007 *)

PARI

x='x+O('x^33);
Vec(serlaplace(exp( x*(1+x)/(1-x) )))
/* Joerg Arndt, Sep 14 2012 */

Formula

E.g.f.: exp(x*(1+x)/(1-x)).

a(n) = Sum_{i=0..n} Sum_{j=0..n} C(i, j)*C(n-j-1, n-i-j)*n!/i!.

D-finite with recurrence: a(n) = (2*n-1)*a(n-1) - (n-4)*(n-1)*a(n-2) - (n-2)*(n-1)*a(n-3). - Vaclav Kotesovec, Jun 27 2013

a(n) ~ 2^(-1/4)*exp(2*sqrt(2*n)-2-n)*n^(n-1/4). - Vaclav Kotesovec, Jun 27 2013