A383603 Expansion of 1/( (1-x)^2 * (1-x-9*x^2) )^(1/3).

1, 1, 4, 7, 28, 67, 250, 703, 2497, 7648, 26488, 85036, 291337, 960769, 3280486, 10993165, 37541611, 127077160, 434707756, 1481346064, 5078811037, 17388735001, 59756049838, 205310507773, 707095964617, 2436104710774, 8406778618336, 29027513057326

Offset: 0

Author

Seiichi Manyama, May 01 2025

Keywords

• nonn

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..400

Crossrefs

Cf. A383597, A383604.

Cf. A383605.

Programs

Magma

R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( 1/( (1-x)^2 * (1-x-9*x^2) )^(1/3))); // Vincenzo Librandi, May 06 2025

Mathematica

CoefficientList[Series[1/((1-x)^2*(1-x-9*x^2))^(1/3),{x,0,27}],x] (* Stefano Spezia, May 02 2025 *)
Table[Sum[(-9)^k*Binomial[-1/3,k]*Binomial[n-k,k],{k,0,Floor[n/2]}],{n,0,30}] (* Vincenzo Librandi, May 06 2025 *)

PARI

a(n) = sum(k=0, n\2, (-9)^k*binomial(-1/3, k)*binomial(n-k, k));

Formula

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

a(n) ~ ((1 + sqrt(37))/2)^(n + 5/3) / (Gamma(1/3) * 3^(4/3) * 37^(1/6) * n^(2/3)). - Vaclav Kotesovec, May 02 2025