A357479 a(n) = (n!/6) * Sum_{k=0..n-3} 1/k!.

0, 0, 0, 1, 8, 50, 320, 2275, 18256, 164388, 1644000, 18084165, 217010200, 2821132886, 39495860768, 592437911975, 9479006592160, 161143112067400, 2900576017214016, 55110944327067273, 1102218886541346600, 23146596617368279930, 509225125582102160000

Offset: 0

Author

Seiichi Manyama, Sep 30 2022

Keywords

• nonn
• easy

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..450

Crossrefs

Column k=3 of A073107.

Cf. A000522, A007526, A038155, A357480.

Cf. A000292, A000449.

Programs

Mathematica

Table[n!/6 Sum[1/k!,{k,0,n-3}],{n,0,30}] (* Harvey P. Dale, Apr 02 2023 *)

PARI

a(n) = n!/6*sum(k=0, n-3, 1/k!);

PARI

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

PARI

my(N=30, x='x+O('x^N)); concat([0, 0, 0], Vec(serlaplace(x^3/6*exp(x)/(1-x))))

PARI

my(N=30, x='x+O('x^N)); concat([0, 0, 0], Vec(sum(k=3, N, k!*x^k/(1-x)^(k+1))/6))

Formula

a(n) = n! * Sum_{k=0..n} binomial(k,3)/k!.

a(0) = 0; a(n) = n * a(n-1) + binomial(n,3).

E.g.f.: x^3/6 * exp(x)/(1-x).

G.f.: (1/6) * Sum_{k>=3} k! * x^k/(1-x)^(k+1).