A383950 Expansion of 1/sqrt((1-2*x)^3 * (1-6*x)).

1, 6, 30, 148, 750, 3924, 21084, 115560, 642582, 3611140, 20455908, 116594328, 667851340, 3840932424, 22164538680, 128269848528, 744150592998, 4326419433060, 25200835078164, 147036927946680, 859181709840804, 5027183713857624, 29450272491511560, 172715082105669552

Offset: 0

Author

Seiichi Manyama, Aug 19 2025

Keywords

• nonn
• changed

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A383948, A383949.

Cf. A231482.

Programs

Magma

R := PowerSeriesRing(Rationals(), 34); f := 1/Sqrt((1- 2*x)^3 * (1-6*x)); coeffs := [ Coefficient(f, n) : n in [0..33] ]; coeffs; // Vincenzo Librandi, Aug 27 2025

Mathematica

CoefficientList[Series[1/Sqrt[(1-2*x)^3*(1-6*x)],{x,0,33}],x] (* Vincenzo Librandi, Aug 27 2025 *)

PARI

my(N=30, x='x+O('x^N)); Vec(1/sqrt((1-2*x)^3*(1-6*x)))

Formula

n*a(n) = (8*n-2)*a(n-1) - 12*n*a(n-2) for n > 1.

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

a(n) = Sum_{k=0..n} (-1)^k * 6^(n-k) * (2*k+1) * binomial(2*k,k) * binomial(n+1,n-k).

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