A351777 Expansion of e.g.f. 1/(1 + 2*x*exp(x)).

1, -2, 4, -6, -8, 150, -972, 3682, 6256, -289746, 3300460, -21071622, -27876312, 3156947014, -53217341660, 494232431250, 175171749088, -113735274256290, 2613309376750812, -32653995355358678, 36013529538641560, 10227377502146048118, -305630239215263764076

Offset: 0

Author

Seiichi Manyama, Feb 19 2022

Keywords

• sign

Crossrefs

Column k=2 of A351776.

Cf. A351762.

Programs

Mathematica

With[{nn=30},CoefficientList[Series[1/(1+2x Exp[x]),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Feb 06 2024 *)

PARI

my(N=40, x='x+O('x^N)); Vec(serlaplace(1/(1+2*x*exp(x))))

PARI

a(n) = n!*sum(k=0, n, (-2)^(n-k)*(n-k)^k/k!);

PARI

a(n) = if(n==0, 1, -2*n*sum(k=0, n-1, binomial(n-1, k)*a(k)));

Formula

a(n) = n! * Sum_{k=0..n} (-2)^(n-k) * (n-k)^k/k!.

a(0) = 1 and a(n) = -2 * n * Sum_{k=0..n-1} binomial(n-1,k) * a(k) for n > 0.