A383602 Expansion of 1/( (1-x) * (1-9*x)^3 )^(1/4).

1, 7, 55, 453, 3819, 32637, 281409, 2441715, 21285411, 186225253, 1633973125, 14370441055, 126631522005, 1117707358515, 9879287145855, 87428272217853, 774533435844531, 6868083093333285, 60952616213098789, 541342619512077967, 4811079933571973329

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, A383600.

Cf. A004982.

Programs

Magma

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

Mathematica

Table[Sum[(-8)^k* Binomial[-3/4,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(-3/4, k)*binomial(n, k));

Formula

a(n) = Sum_{k=0..n} (-8)^k * binomial(-3/4,k) * binomial(n,k).

n*a(n) = (10*n-3)*a(n-1) - 9*(n-1)*a(n-2) for n > 1.

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

a(n) = hypergeom([3/4, -n], [1], -8). - Stefano Spezia, May 05 2025