A387282 - OEIS

%I #17 Aug 26 2025 14:22:08

%S 1,36,738,11352,145926,1657656,17202900,166651056,1529421894,

%T 13438354072,113934017340,937605593808,7523844806556,59086320919344,

%U 455434002675432,3453696244883808,25817301841465926,190551351969051480,1390535665902059820,10044442002527721360

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

%H Vincenzo Librandi, <a href="/A387282/b387282.txt">Table of n, a(n) for n = 0..500</a>

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

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

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

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

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

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

%t CoefficientList[Series[1/((1-2x)*(1-6*x))^(9/2),{x,0,33}],x] (* _Vincenzo Librandi_, Aug 26 2025 *)

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

%o (Magma) R<x> := 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

%Y Cf. A387274.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Aug 24 2025