A387285 - OEIS

%I #22 Aug 26 2025 08:37:45

%S 1,45,1143,21615,339273,4678245,58635291,683043075,7509979620,

%T 78804997700,795761196516,7781553124020,74052668067768,

%U 688495224545640,6273531262685808,56167899323785920,495164048016313071,4305882857891533875,36989093846561967825,314291267108824487625

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

%H Paolo Xausa, <a href="/A387285/b387285.txt">Table of n, a(n) for n = 0..1000</a>

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

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

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

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

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

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

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

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

%o (Magma) R<x> := 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

%Y Cf. A387277.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Aug 24 2025