A383605 Expansion of 1/( (1-x) * (1-x-9*x^2)^2 )^(1/3).

1, 1, 7, 13, 64, 160, 661, 1927, 7288, 23044, 83413, 275479, 976198, 3301462, 11584861, 39703783, 138747637, 479200129, 1672353256, 5803085008, 20251472416, 70486033288, 246114881956, 858397066324, 2999541427177, 10477699520329, 36642516789607, 128146441442989

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, A383606.

Programs

Magma

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

Mathematica

CoefficientList[Series[1/((1-x)*(1-x-9*x^2)^2)^(1/3),{x,0,27}],x] (* Stefano Spezia, May 02 2025 *)
Table[Sum[(-9)^k*Binomial[-2/3,k]*Binomial[n-k,k],{k,0,Floor[n/2]}],{n,0,35}] (* Vincenzo Librandi, May 06 2025 *)

PARI

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

Formula

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

a(n) ~ Gamma(1/3) * (1 + sqrt(37))^(n + 4/3) / (Pi * 3^(1/6) * 37^(1/3) * n^(1/3) * 2^(n + 7/3)). - Vaclav Kotesovec, May 02 2025