A391876 a(n) = Sum_{k=0..n} 2^(n-k) * binomial(k+2,2) * binomial(2*k,2*(n-k)).

1, 3, 12, 82, 339, 1461, 6198, 24732, 97557, 377095, 1431792, 5370102, 19904711, 73046121, 265779354, 959669856, 3441865065, 12270185691, 43506695476, 153510593754, 539253893883, 1886647300701, 6576263406270, 22844885842020, 79110694279549, 273161800735983, 940663666431288

Offset

0,2

Links

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

Index entries for linear recurrences with constant coefficients, signature (6,-3,-16,-51,78,147,156,-204,-128,-48,192,-64).

Formula

G.f.: (1-x-2*x^2) * ((1-x-2*x^2)^2 + 24*x^3)/((1-x-2*x^2)^2 - 8*x^3)^3.

a(n) = 6*a(n-1) - 3*a(n-2) - 16*a(n-3) - 51*a(n-4) + 78*a(n-5) + 147*a(n-6) + 156*a(n-7) - 204*a(n-8) - 128*a(n-9) - 48*a(n-10) + 192*a(n-11) - 64*a(n-12).

Mathematica

CoefficientList[Series[(1-x-2*x^2)*((1-x-2*x^2)^2+24*x^3)/((1-x-2*x^2)^2-8*x^3)^3, {x, 0, 30}], x] (* Vincenzo Librandi, Jan 02 2026 *)

Prog

(PARI) my(A=1, B=2, 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-x-2*x^2) * ((1-x-2*x^2)^2 + 24*x^3)/((1-x-2*x^2)^2 - 8*x^3)^3); // Vincenzo Librandi, Jan 02 2026

Crossrefs

Cf. A108480, A390732.

Sequence in context: A188227 A357665 A224608 * A023881 A381570 A067111

Adjacent sequences: A391873 A391874 A391875 * A391877 A391878 A391879

Keyword

nonn,easy

Author

Seiichi Manyama, Dec 21 2025

Status

approved