A375411 Expansion of e.g.f. exp(-x * (1 - x)^3) / (1 - x).

1, 0, 7, -16, 177, -1096, 10975, -94872, 1101121, -11699632, 151701111, -1897734400, 27287272177, -385421578296, 6100570870927, -95315920570696, 1642003509857025, -27968228816277472, 520462884927746791, -9551232423922438512, 190797743531054785201

Offset: 0

Author

Seiichi Manyama, Aug 14 2024

Keywords

• sign

Crossrefs

Cf. A375409, A375410.

Programs

PARI

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

PARI

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

Formula

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

D-finite with recurrence a(n) +(-n+1)*a(n-1) +7*(-n+1)*a(n-2) +15*(n-1)*(n-2)*a(n-3) -13*(n-1)*(n-2)*(n-3)*a(n-4) +4*(n-1)*(n-2)*(n-3)*(n-4)*a(n-5)=0. - R. J. Mathar, Aug 14 2024