A351778 Expansion of e.g.f. 1/(1 + 3*x*exp(x)).

1, -3, 12, -63, 420, -3435, 33462, -379155, 4903896, -71318259, 1152202290, -20474486043, 396890715636, -8334602179995, 188486823883134, -4567087352339235, 118039115079323952, -3241465018561379427, 94249758656850366186, -2892678859033260044043

Offset: 0

Author

Seiichi Manyama, Feb 19 2022

Keywords

• sign

Crossrefs

Column k=3 of A351776.

Cf. A351763.

Programs

Mathematica

With[{nn=30},CoefficientList[Series[1/(1+3x Exp[x]),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, May 21 2024 *)

PARI

my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(1+3*x*exp(x))))

PARI

a(n) = n!*sum(k=0, n, (-3)^(n-k)*(n-k)^k/k!);

PARI

a(n) = if(n==0, 1, -3*n*sum(k=0, n-1, binomial(n-1, k)*a(k)));

Formula

a(n) = n! * Sum_{k=0..n} (-3)^(n-k) * (n-k)^k/k!.

a(0) = 1 and a(n) = -3 * n * Sum_{k=0..n-1} binomial(n-1,k) * a(k) for n > 0.