A391875 - OEIS

A391875

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

1, 6, 30, 224, 1464, 8592, 48912, 270720, 1459248, 7709216, 40068000, 205378560, 1040357120, 5216715264, 25927942656, 127867834368, 626272781568, 3048584609280, 14758392452608, 71091792715776, 340910500030464, 1628080470994944, 7746005007544320, 36726482563399680, 173579373208170496

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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,-48,88,-204,480,-384,960,-816,704,-768,384,-64).

FORMULA

G.f.: (1-2*x-2*x^2) * ((1-2*x-2*x^2)^2 + 48*x^3)/((1-2*x-2*x^2)^2 - 16*x^3)^3.

a(n) = 12*a(n-1) - 48*a(n-2) + 88*a(n-3) - 204*a(n-4) + 480*a(n-5) - 384*a(n-6) + 960*a(n-7) - 816*a(n-8) + 704*a(n-9) - 768*a(n-10) + 384*a(n-11) - 64*a(n-12).

MATHEMATICA

CoefficientList[Series[(1-2*x-2*x^2)*((1-2*x-2*x^2)^2+48*x^3)/((1-2*x-2*x^2)^2-16*x^3)^3, {x, 0, 30}], x] (* Vincenzo Librandi, Jan 02 2026 *)

PROG

(PARI) my(A=2, B=1, C=4*A^2*B, N=3, 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)

(Magma) m:=30; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R! (1-2*x-2*x^2) * ((1-2*x-2*x^2)^2 + 48*x^3)/((1-2*x-2*x^2)^2 - 16*x^3)^3); // Vincenzo Librandi, Jan 02 2026

CROSSREFS

Cf. A108485, A390700.

Sequence in context: A298457 A208942 A209070 * A259820 A126751 A009689

Adjacent sequences: A391872 A391873 A391874 * A391876 A391877 A391878

KEYWORD

nonn,easy

AUTHOR

Seiichi Manyama, Dec 21 2025

STATUS

approved