A383951 Expansion of 1/sqrt((1-x)^3 * (1-13*x)).

1, 8, 75, 796, 8975, 104532, 1242241, 14967680, 182172627, 2234388520, 27572562017, 341929238196, 4257601409425, 53196292831052, 666600870098895, 8374220026541296, 105432936550339523, 1329984626694890760, 16805828389675759921, 212684693606424187460, 2695304533342226489229

Offset: 0

Author

Seiichi Manyama, Aug 19 2025

Keywords

• nonn
• changed

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..800

Crossrefs

Partial sums of A340973.

Programs

Magma

R := PowerSeriesRing(Rationals(), 34); f := 1/Sqrt((1- x)^3 * (1-13*x)); coeffs := [ Coefficient(f, n) : n in [0..33] ]; coeffs; // Vincenzo Librandi, Aug 27 2025

Mathematica

CoefficientList[Series[1/Sqrt[(1-x)^3*(1-13*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*(1-13*x)))

Formula

n*a(n) = (14*n-6)*a(n-1) - 13*n*a(n-2) for n > 1.

a(n) = (1/4)^n * Sum_{k=0..n} 13^(n-k) * (2*k+1) * binomial(2*k,k) * binomial(2*(n-k),n-k).

a(n) = Sum_{k=0..n} (-3)^k * 13^(n-k) * (2*k+1) * binomial(2*k,k) * binomial(n+1,n-k).

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