A387285 Expansion of 1/((1-3*x) * (1-7*x))^(9/2).

1, 45, 1143, 21615, 339273, 4678245, 58635291, 683043075, 7509979620, 78804997700, 795761196516, 7781553124020, 74052668067768, 688495224545640, 6273531262685808, 56167899323785920, 495164048016313071, 4305882857891533875, 36989093846561967825, 314291267108824487625

Offset: 0

Author

Seiichi Manyama, Aug 24 2025

Keywords

• nonn
• new

Links

• Paolo Xausa, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A387277.

Programs

Magma

R := PowerSeriesRing(Rationals(), 34); f := 1/((1-3*x) * (1-7*x))^(9/2); coeffs := [ Coefficient(f, n) : n in [0..33] ]; coeffs; // Vincenzo Librandi, Aug 26 2025

Mathematica

Module[{x}, CoefficientList[Series[1/((1 - 3*x)*(1 - 7*x))^(9/2), {x, 0, 25}], x]] (* Paolo Xausa, Aug 25 2025 *)

PARI

my(N=20, x='x+O('x^N)); Vec(1/((1-3*x)*(1-7*x))^(9/2))

Formula

n*a(n) = (10*n+35)*a(n-1) - 21*(n+7)*a(n-2) for n > 1.

a(n) = (-1)^n * Sum_{k=0..n} 7^k * 3^(n-k) * binomial(-9/2,k) * binomial(-9/2,n-k).

a(n) = Sum_{k=0..n} (-4)^k * 3^(n-k) * binomial(-9/2,k) * binomial(n+8,n-k).

a(n) = Sum_{k=0..n} 4^k * 7^(n-k) * binomial(-9/2,k) * binomial(n+8,n-k).

a(n) = (binomial(n+8,4)/70) * A387277(n).

a(n) = (-1)^n * Sum_{k=0..n} 10^k * (21/10)^(n-k) * binomial(-9/2,k) * binomial(k,n-k).