A387282 Expansion of 1/((1-2*x) * (1-6*x))^(9/2).

1, 36, 738, 11352, 145926, 1657656, 17202900, 166651056, 1529421894, 13438354072, 113934017340, 937605593808, 7523844806556, 59086320919344, 455434002675432, 3453696244883808, 25817301841465926, 190551351969051480, 1390535665902059820, 10044442002527721360

Offset: 0

Author

Seiichi Manyama, Aug 24 2025

Keywords

• nonn
• new

Links

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

Crossrefs

Cf. A387274.

Programs

Magma

R := PowerSeriesRing(Rationals(), 34); f := 1/((1-2*x) * (1-6*x))^(9/2); coeffs := [ Coefficient(f, n) : n in [0..33] ]; coeffs; // Vincenzo Librandi, Aug 26 2025

Mathematica

CoefficientList[Series[1/((1-2x)*(1-6*x))^(9/2),{x,0,33}],x] (* Vincenzo Librandi, Aug 26 2025 *)

PARI

my(N=20, x='x+O('x^N)); Vec(1/((1-2*x)*(1-6*x))^(9/2))

Formula

n*a(n) = (8*n+28)*a(n-1) - 12*(n+7)*a(n-2) for n > 1.

a(n) = (-2)^n * Sum_{k=0..n} 3^k * binomial(-9/2,k) * binomial(-9/2,n-k).

a(n) = 2^n * Sum_{k=0..n} (-2)^k * binomial(-9/2,k) * binomial(n+8,n-k).

a(n) = Sum_{k=0..n} 4^k * 6^(n-k) * binomial(-9/2,k) * binomial(n+8,n-k).

a(n) = (binomial(n+8,4)/70) * A387274(n).

a(n) = (-1)^n * Sum_{k=0..n} 8^k * (3/2)^(n-k) * binomial(-9/2,k) * binomial(k,n-k).