A375410 Expansion of e.g.f. exp(-x * (1 - x)^2) / (1 - x).

1, 0, 5, -4, 81, -176, 2605, -8100, 137249, -424576, 10376181, -21429860, 1069514545, 279470736, 149969551901, 616166705084, 28838719110465, 261581059999360, 7560615053166949, 106911086586605244, 2626348956282622481, 48474495094075756880, 1160413567193463596685

Offset: 0

Author

Seiichi Manyama, Aug 14 2024

Keywords

• sign

Crossrefs

Cf. A375409, A375411.

Cf. A375413, A375415.

Programs

PARI

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(-x*(1-x)^2)/(1-x)))

PARI

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

Formula

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

D-finite with recurrence a(n) +(-n+1)*a(n-1) +5*(-n+1)*a(n-2) +7*(n-1)*(n-2)*a(n-3) -3*(n-1)*(n-2)*(n-3)*a(n-4)=0. - R. J. Mathar, Aug 14 2024