A380309 Expansion of e.g.f. exp( 1 - 1/(1-3*x)^(1/3) ).

1, -1, -3, -17, -143, -1601, -22419, -377217, -7415743, -166854657, -4229195779, -119251176881, -3702809175823, -125546570425537, -4615357640315603, -182855338776726561, -7766868454872857599, -352082642456714366977, -16965451818345573907843

Offset: 0

Author

Seiichi Manyama, Jan 20 2025

Keywords

• sign

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..382

Crossrefs

Cf. A145561, A380308, A380310.

Cf. A000587, A007559, A015735.

Programs

PARI

my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(1-1/(1-3*x)^(1/3))))

Formula

a(n) = Sum_{k=0..n} 3^(n-k) * |Stirling1(n,k)| * A000587(k).

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

a(0) = 1; a(n) = -Sum_{k=1..n} A007559(k) * binomial(n-1,k-1) * a(n-k).