A372024 Expansion of ( (1 + 8*x)/(1 - x) )^(1/3).

1, 3, -6, 30, -159, 939, -5856, 37992, -253590, 1729758, -12001524, 84422244, -600613998, 4313760870, -31233564312, 227721316008, -1670358339735, 12317389232475, -91256834358510, 678934503248070, -5070123478493727, 37990712430826059, -285540106663543128

Offset: 0

Author

Seiichi Manyama, Apr 16 2024

Keywords

• sign

Crossrefs

Cf. A372023.

Programs

PARI

my(N=30, x='x+O('x^N)); Vec(((1+8*x)/(1-x))^(1/3))

PARI

a(n) = sum(k=0, n, 9^k*binomial(1/3, k)*binomial(n-1, n-k));

Formula

a(n) = Sum_{k=0..n} 9^k * binomial(1/3,k) * binomial(n-1,n-k).

a(n) ~ (-1)^(n+1) * 2^(3*n) * Gamma(1/3) / (Pi * 3^(7/6) * n^(4/3)). - Vaclav Kotesovec, Apr 16 2024

D-finite with recurrence n*a(n) +(7*n-10)*a(n-1) +8*(-n+2)*a(n-2)=0. - R. J. Mathar, Apr 22 2024