A370670 Expansion of Sum_{k>=0} k! * ( x/(1+x^3) )^k.

1, 1, 2, 6, 23, 116, 702, 4945, 39726, 358596, 3593759, 39596032, 475750740, 6190873441, 86740653730, 1301942638170, 20842037779079, 354469561697988, 6382795892548194, 121310901632237857, 2426864464216669694, 50975856191753357928, 1121692313538562441535

Offset: 0

Author

Seiichi Manyama, Feb 25 2024

Keywords

• nonn

Crossrefs

Cf. A000255, A370669.

Cf. A177251, A370508, A370511.

Programs

PARI

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

PARI

a(n) = sum(k=0, n\3, (-1)^k*(n-3*k)!*binomial(n-2*k-1, k));

Formula

a(n) = Sum_{k=0..floor(n/3)} (-1)^k * (n-3*k)! * binomial(n-2*k-1,k).

a(n) = n*a(n-1) + a(n-3) + (n-6)*a(n-4) + 2*a(n-6) for n > 6.