A391965 - OEIS

%I #17 Dec 30 2025 11:42:04

%S 1,6,24,104,480,2112,8752,34848,135168,513792,1920000,7071744,

%T 25728256,92622336,330405888,1169217536,4108296192,14344470528,

%U 49802211328,172027060224,591476293632,2025119416320,6907067105280,23474916556800,79524883136512,268595063291904

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

%H Seiichi Manyama, <a href="/A391965/b391965.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (12,-60,172,-336,480,-496,384,-192,64).

%F G.f.: (1-2*x) * ((1-2*x)^2 + 12*x^3) / ((1-2*x)^2 - 4*x^3)^3.

%F a(n) = 12*a(n-1) - 60*a(n-2) + 172*a(n-3) - 336*a(n-4) + 480*a(n-5) - 496*a(n-6) + 384*a(n-7) - 192*a(n-8) + 64*a(n-9).

%t CoefficientList[Series[(1-2*x)*((1-2*x)^2+12*x^3)/((1-2*x)^2-4*x^3)^3,{x,0,50}],x] (* _Vincenzo Librandi_, Dec 30 2025 *)

%o (PARI) my(A=2, B=1, C=A^2*B, N=3, M=30, x='x+O('x^M), X=1-A*x, Y=3); Vec(sum(k=0, N\2, C^k*binomial(N, 2*k)*X^(N-2*k)*x^(Y*k))/(X^2-C*x^Y)^N)

%o (Magma) m:=50; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!((1-2*x) * ((1-2*x)^2 + 12*x^3) / ((1-2*x)^2 - 4*x^3)^3)); // _Vincenzo Librandi_, Dec 30 2025

%Y Cf. A391875.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Dec 23 2025