A390775 - OEIS

%I #38 Dec 29 2025 12:30:26

%S 1,0,0,2,0,0,4,12,0,8,72,0,16,288,144,32,960,1440,64,2880,8640,1856,

%T 8064,40320,24448,21504,161280,194048,76032,580608,1162240,511488,

%U 1935360,5808128,4070400,6331392,25550848,28182528,23721984,102195200,166146048

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

%H Seiichi Manyama, <a href="/A390775/b390775.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,4,0,0,-4,12).

%F G.f.: (1-2*x^3) / ((1-2*x^3)^2 - 12*x^7).

%F a(n) = 4*a(n-3) - 4*a(n-6) + 12*a(n-7).

%t CoefficientList[Series[(1-2*x^3)/((1-2*x^3)^2-12*x^7),{x,0,50}],x] (* _Vincenzo Librandi_, Dec 29 2025 *)

%o (PARI) my(A=2, B=3, C=A^2*B, N=1, M=50, x='x+O('x^M), X=1-A*x^3, Y=7); 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^3)/((1-2*x^3)^2-12*x^7)); // _Vincenzo Librandi_, Dec 29 2025

%Y Cf. A391960, A391961.

%Y Cf. A390619.

%K nonn

%O 0,4

%A _Seiichi Manyama_, Dec 23 2025