A356814 a(n) = Sum_{k=0..n} (-1)^k * (k*n+1)^(n-k) * binomial(n,k).

1, 0, -4, -27, -64, 4375, 199584, 6739607, 169934848, -1012395105, -709624000000, -86599643309201, -8221227668471808, -638169258399740977, -27617164284655812608, 3853095093357099609375, 1568756883209662050074624, 360407172063462944082773311

Offset: 0

Author

Seiichi Manyama, Aug 29 2022

Keywords

• sign

Crossrefs

Cf. A292893, A356812, A356813.

Cf. A320258, A356806, A356811, A356817.

Programs

PARI

a(n) = sum(k=0, n, (-1)^k*(k*n+1)^(n-k)*binomial(n, k));

PARI

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

Formula

a(n) = n! * [x^n] exp( x * (1 - exp(n * x)) ).

a(n) = [x^n] Sum_{k>=0} (-x)^k / (1 - (n*k+1)*x)^(k+1).

a(n) = n! * Sum_{k=0..floor(n/2)} (-1)^k * n^(n-k) * Stirling2(n-k,k)/(n-k)!.