A383597 Expansion of 1/( (1-x)^2 * (1-10*x) )^(1/3).

1, 4, 25, 190, 1570, 13552, 120178, 1085620, 9940345, 91962460, 857750233, 8053389142, 76026759760, 721017894640, 6864725124520, 65578937628304, 628320730656586, 6035594205744520, 58110220504754650, 560624083417180300, 5418599393597801020, 52459116546784350880

Offset

0,2

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..300

Formula

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

n*a(n) = (11*n-7)*a(n-1) - 10*(n-1)*a(n-2) for n > 1.

a(n) ~ 10^(n + 2/3) / (Gamma(1/3) * 3^(4/3) * n^(2/3)). - Vaclav Kotesovec, May 02 2025

a(n) = hypergeom([1/3, -n], [1], -9). - Stefano Spezia, May 04 2025

Mathematica

Table[Sum[(-9)^k *Binomial[-1/3, k]* Binomial[n, k], {k, 0, n}], {n, 0, 25}] (* Vincenzo Librandi, May 04 2025 *)

Prog

(PARI) a(n) = sum(k=0, n, (-9)^k*binomial(-1/3, k)*binomial(n, k));
(Magma) I:=[4, 25]; [1] cat [n le 2 select I[n] else ((11*n-7)*Self(n-1) - 10*(n-1) *Self(n-2))/n : n in [1..30]]; // Vincenzo Librandi, May 04 2025

Crossrefs

Cf. A383598, A383599.

Cf. A004987, A361375, A376802, A383601.

Sequence in context: A215791 A175913 A239996 * A064063 A171991 A380678

Adjacent sequences: A383594 A383595 A383596 * A383598 A383599 A383600

Keyword

nonn,easy

Author

Seiichi Manyama, May 01 2025

Status

approved