A391965 a(n) = Sum_{k=0..n} 2^k * binomial(k+2,2) * binomial(k,2*(n-k)).

1, 6, 24, 104, 480, 2112, 8752, 34848, 135168, 513792, 1920000, 7071744, 25728256, 92622336, 330405888, 1169217536, 4108296192, 14344470528, 49802211328, 172027060224, 591476293632, 2025119416320, 6907067105280, 23474916556800, 79524883136512, 268595063291904

Offset

0,2

Links

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (12,-60,172,-336,480,-496,384,-192,64).

Formula

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

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).

Mathematica

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 *)

Prog

(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)
(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

Crossrefs

Cf. A391875.

Sequence in context: A120583 A089378 A074414 * A165793 A344274 A372225

Adjacent sequences: A391962 A391963 A391964 * A391966 A391967 A391968

Keyword

nonn

Author

Seiichi Manyama, Dec 23 2025

Status

approved