A383600 Expansion of 1/( (1-x)^3 * (1-9*x) )^(1/4).

1, 3, 15, 97, 699, 5313, 41689, 334215, 2721411, 22423737, 186497325, 1562826195, 13178010405, 111700773135, 951026829255, 8128169277897, 69701329848051, 599462375836185, 5169038197383789, 44674793959777443, 386916485124220929, 3357265884164614707

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. A084771, A383602.

Cf. A004981, A231482.

Programs

Magma

R:=PowerSeriesRing(Rationals(), 25); Coefficients(R!( 1/( (1-x)^3 * (1-9*x) )^(1/4))); // Vincenzo Librandi, May 05 2025

Mathematica

Table[Sum[(-8)^(k)* Binomial[-1/4,k]* Binomial[n,k],{k,0,n}],{n,0,22}] (* Vincenzo Librandi, May 05 2025 *)

PARI

a(n) = sum(k=0, n, (-8)^k*binomial(-1/4, k)*binomial(n, k));

Formula

a(n) = Sum_{k=0..n} (-8)^k * binomial(-1/4,k) * binomial(n,k).

n*a(n) = (10*n-7)*a(n-1) - 9*(n-1)*a(n-2) for n > 1.

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

a(n) = hypergeom([1/4, -n], [1], -8). - Stefano Spezia, May 05 2025