A383601 Expansion of 1/( (1-x) * (1-10*x)^2 )^(1/3).

1, 7, 58, 514, 4705, 43879, 414208, 3943492, 37782346, 363760390, 3515819020, 34088616940, 331383573010, 3228590970430, 31514912933800, 308126549765440, 3016908101224105, 29576113797737695, 290271761086278610, 2851684765215491050, 28040613734007656545

Offset: 0

Author

Seiichi Manyama, May 01 2025

Keywords

• nonn
• easy

Links

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

Crossrefs

Cf. A383605, A383606.

Cf. A004988, A377233, A383597.

Programs

Magma

R:=PowerSeriesRing(Rationals(), 25); Coefficients(R!( 1/( (1-x) * (1-10*x)^2 )^(1/3))); // Vincenzo Librandi, May 05 2025

Mathematica

Table[Sum[(-9)^k* Binomial[-2/3,k]* Binomial[n,k],{k,0,n}],{n,0,22}] (* Vincenzo Librandi, May 05 2025 *)

PARI

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

Formula

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

n*a(n) = (11*n-4)*a(n-1) - 10*(n-1)*a(n-2) for n > 1.

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

a(n) = hypergeom([2/3, -n], [1], -9). - Stefano Spezia, May 04 2025