A383598 Expansion of 1/( (1-x^2)^2 * (1-x^2-9*x) )^(1/3).

1, 3, 19, 132, 1000, 7884, 63802, 525666, 4388518, 37010220, 314633944, 2692239012, 23161121641, 200158043223, 1736461678195, 15114944308560, 131950690469920, 1154858014686960, 10130508263000440, 89045875688728440, 784127521246844872, 6916291864328172336

Offset

0,2

Links

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

Formula

a(n) = Sum_{k=0..floor(n/2)} (-9)^(n-2*k) * binomial(-1/3,n-2*k) * binomial(n-k,k).

a(n) ~ ((9 + sqrt(85))/2)^(n+1) / (Gamma(1/3) * 3^(4/3) * 85^(1/6) * n^(2/3)). - Vaclav Kotesovec, May 02 2025

Mathematica

Table[Sum[(-9)^(n-2*k)* Binomial[-1/3, n-2*k]* Binomial[n-k, k], {k, 0, Floor[n/2]}], {n, 0, 22}] (* Vincenzo Librandi, May 04 2025 *)

Prog

(PARI) a(n) = sum(k=0, n\2, (-9)^(n-2*k)*binomial(-1/3, n-2*k)*binomial(n-k, k));
(Magma) R<x>:=PowerSeriesRing(Rationals(), 25); Coefficients(R!( 1/( (1-x^2)^2 * (1-x^2-9*x) )^(1/3))); // Vincenzo Librandi, May 04 2025

Crossrefs

Cf. A383597, A383599.

Cf. A376805.

Sequence in context: A256092 A370159 A369627 * A074567 A332621 A091346

Adjacent sequences: A383595 A383596 A383597 * A383599 A383600 A383601

Keyword

nonn

Author

Seiichi Manyama, May 01 2025

Status

approved