A360987 E.g.f. A(x) satisfies A(x) = exp(x * A(-x)^2).

1, 1, -3, -23, 233, 3521, -62171, -1416407, 35880977, 1095318721, -36224195059, -1387587617239, 56675849155705, 2612993427672577, -127090039302776395, -6852033608852338199, 386750643197222855969, 23875394847093826450049

Offset: 0

Author

Seiichi Manyama, Feb 27 2023

Keywords

• sign

Comments

Sum_{k=0..n} (2*n - 2*k + 1)^(k-1) * (2*k)^(n-k) * binomial(n,k) = (2*n+1)^(n-1) = A052750(n). - Vaclav Kotesovec, Jul 03 2025

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..367

Crossrefs

Cf. A052750, A141369, A196198, A360988, A360989, A360990.

Cf. A168600.

Programs

PARI

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

Formula

a(n) = Sum_{k=0..n} (2*n - 2*k + 1)^(k-1) * (-2*k)^(n-k) * binomial(n,k).

a(0) = 1; a(n) = (-1)^(n-1) * (n-1)! * Sum_{i, j, k>=0 and i+j+k=n-1} (-1)^i * (n-i) * a(i) * a(j) * a(k)/(i! * j! * k!). - Seiichi Manyama, Jul 06 2025