A390732 - OEIS

%I #27 Jan 01 2026 15:16:16

%S 1,2,7,40,137,526,1979,7092,25485,90266,316415,1102336,3813841,

%T 13123366,44947139,153295692,520920213,1764401586,5958744327,

%U 20071157144,67446281689,226154549822,756826028235,2528141873892,8431100881821,28073731414730,93346571969551,309972436014384

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

%H Seiichi Manyama, <a href="/A390732/b390732.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (4,2,-4,-33,-8,8,32,-16).

%F G.f.: ((1-x-2*x^2)^2 + 8*x^3)/((1-x-2*x^2)^2 - 8*x^3)^2.

%F a(n) = 4*a(n-1) + 2*a(n-2) - 4*a(n-3) - 33*a(n-4) - 8*a(n-5) + 8*a(n-6) + 32*a(n-7) - 16*a(n-8).

%t CoefficientList[Series[((1-x-2*x^2)^2+8*x^3)/((1-x-2*x^2)^2-8*x^3)^2,{x,0,50}],x] (* _Vincenzo Librandi_, Jan 01 2026 *)

%o (PARI) my(A=1, B=2, C=4*A^2*B, N=2, M=30, x='x+O('x^M), X=1-A*x-A*B*x^2, 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-x-2*x^2)^2 + 8*x^3)/((1-x-2*x^2)^2 - 8*x^3)^2); // _Vincenzo Librandi_, Jan 01 2026

%Y Cf. A108480, A391876.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Dec 21 2025