A383948 Expansion of 1/sqrt((1-3*x)^3 * (1-7*x)).

1, 8, 51, 308, 1855, 11340, 70665, 448320, 2887155, 18815240, 123759097, 819969276, 5464090177, 36580917716, 245837438055, 1657396783440, 11204207037315, 75918595916520, 515462211835305, 3506072423912940, 23885410548196701, 162951783575205108, 1113110415733083531

Offset: 0

Author

Seiichi Manyama, Aug 19 2025

Keywords

• nonn
• changed

Links

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

Crossrefs

Cf. A098409, A360318.

Cf. A383949, A383950.

Cf. A132894.

Programs

Magma

R := PowerSeriesRing(Rationals(), 34); f := 1/Sqrt((1- 3*x)^3 * (1-7*x)); coeffs := [ Coefficient(f, n) : n in [0..33] ]; coeffs; // Vincenzo Librandi, Aug 27 2025

Mathematica

CoefficientList[Series[ 1/Sqrt[(1-3*x)^3*(1-7*x)],{x,0,33}],x] (* Vincenzo Librandi, Aug 27 2025 *)

PARI

my(N=30, x='x+O('x^N)); Vec(1/sqrt((1-3*x)^3*(1-7*x)))

Formula

n*a(n) = (10*n-2)*a(n-1) - 21*n*a(n-2) for n > 1.

a(n) = (1/4)^n * Sum_{k=0..n} 3^k * 7^(n-k) * (2*k+1) * binomial(2*k,k) * binomial(2*(n-k),n-k).

a(n) = Sum_{k=0..n} (-1)^k * 7^(n-k) * (2*k+1) * binomial(2*k,k) * binomial(n+1,n-k).

a(n) = Sum_{k=0..n} 3^(n-k) * binomial(2*k,k) * binomial(n+1,n-k).