A383949 Expansion of 1/sqrt((1-x)^3 * (1-5*x)).

1, 4, 15, 60, 255, 1128, 5117, 23600, 110115, 518220, 2455101, 11693124, 55934385, 268535400, 1293178275, 6243968880, 30217425795, 146529719100, 711810105725, 3463284659300, 16874328961245, 82322471522280, 402079323279975, 1965900162652800, 9621179345962525, 47127880914834148

Offset: 0

Author

Seiichi Manyama, Aug 19 2025

Keywords

• nonn
• new

Links

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

Crossrefs

Cf. A383948, A383950.

Cf. A052244.

Programs

Magma

R := PowerSeriesRing(Rationals(), 34); f := 1/Sqrt((1- x)^3 * (1-5*x)); coeffs := [ Coefficient(f, n) : n in [0..33] ]; coeffs; // Vincenzo Librandi, Aug 27 2025

Mathematica

CoefficientList[Series[1/Sqrt[(1-x)^3*(1-5*x)],{x,0,33}],x] (* Vincenzo Librandi, Aug 27 2025 *)

PARI

my(N=30, x='x+O('x^N)); Vec(1/sqrt((1-x)^3*(1-5*x)))

Formula

n*a(n) = (6*n-2)*a(n-1) - 5*n*a(n-2) for n > 1.

a(n) = (1/4)^n * Sum_{k=0..n} 5^(n-k) * (2*k+1) * binomial(2*k,k) * binomial(2*(n-k),n-k).

a(n) = Sum_{k=0..n} (-1)^k * 5^(n-k) * (2*k+1) * binomial(2*k,k) * binomial(n+1,n-k).

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