A383606 Expansion of 1/( (1-x) * (1-x-9*x^3)^2 )^(1/3).

1, 1, 1, 7, 13, 19, 70, 166, 307, 853, 2164, 4600, 11491, 29137, 66808, 161692, 403843, 961129, 2316238, 5715742, 13831219, 33450073, 82013692, 199820584, 485389276, 1187152906, 2900334583, 7069398325, 17283884710, 42278723290, 103291322056, 252668924536

Offset: 0

Author

Seiichi Manyama, May 01 2025

Keywords

• nonn

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..500

Crossrefs

Cf. A383601, A383605.

Programs

Magma

R:=PowerSeriesRing(Rationals(), 35); Coefficients(R!( 1/( (1-x) * (1-x-9*x^3)^2 )^(1/3))); // Vincenzo Librandi, May 06 2025

Mathematica

CoefficientList[Series[1/((1-x)*(1-x-9*x^3)^2)^(1/3),{x,0,31}],x] (* Stefano Spezia, May 02 2025 *)
Table[Sum[(-9)^k*Binomial[-2/3,k]*Binomial[n-2*k,k],{k,0,Floor[n/3]}],{n,0,35}] (* Vincenzo Librandi, May 06 2025 *)

PARI

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

Formula

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