A391964 - OEIS

%I #16 Dec 30 2025 11:42:00

%S 1,4,12,44,176,672,2448,8704,30528,105920,364032,1241088,4202752,

%T 14150656,47410176,158157824,525602816,1740840960,5748461568,

%U 18930761728,62190206976,203850235904,666838499328,2177331363840,7097215418368,23097693700096,75061550383104

%N a(n) = Sum_{k=0..n} (k+1) * 2^k * binomial(k,2*(n-k)).

%H Seiichi Manyama, <a href="/A391964/b391964.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (8,-24,40,-48,32,-16).

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

%F a(n) = 8*a(n-1) - 24*a(n-2) + 40*a(n-3) - 48*a(n-4) + 32*a(n-5) - 16*a(n-6).

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

%o (PARI) my(A=2, B=1, C=A^2*B, N=2, 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)^2 + 4*x^3) / ((1-2*x)^2 - 4*x^3)^2); // _Vincenzo Librandi_, Dec 30 2025

%Y Cf. A390700.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Dec 23 2025