A368237 Expansion of e.g.f. 1/(exp(-x) - 3*x).

1, 4, 31, 361, 5605, 108781, 2533447, 68836279, 2137543177, 74673228457, 2898494302651, 123757822391083, 5764497138070381, 290878956151681405, 15806942065094830735, 920336494043393536591, 57157621592164505969425, 3771643127452655490322513

Offset: 0

Author

Seiichi Manyama, Dec 18 2023

Keywords

• nonn

Crossrefs

Cf. A072597, A368236.

Cf. A336948.

Programs

Mathematica

a[n_] := n! Sum[3^(n - k) (n - k + 1)^k / k!, {k, 0, n}];Table[a[n],{n,0,17}] (* or *) a[0] = 1;a[n_] := 3n a[n - 1] + Sum[(-1)^(k - 1) Binomial[n, k] a[n - k], {k, 1, n}];Table[a[n],{n,0,17}] (* James C. McMahon, Dec 18 2023 *)

PARI

a(n) = n!*sum(k=0, n, 3^(n-k)*(n-k+1)^k/k!);

PARI

my(x='x+O('x^25)); Vec(serlaplace(1/(exp(-x) - 3*x))) \\ Michel Marcus, Dec 18 2023

Formula

a(0) = 1; a(n) = 3*n*a(n-1) + Sum_{k=1..n} (-1)^(k-1) * binomial(n,k) * a(n-k).

a(n) = n! * Sum_{k=0..n} 3^(n-k) * (n-k+1)^k / k!.