A392269 a(n) = Sum_{k=0..floor(2*n/3)} binomial(k+2,2) * binomial(k,2*n-3*k).

1, 0, 3, 6, 6, 30, 25, 90, 120, 238, 441, 672, 1333, 2016, 3681, 5946, 9945, 16698, 26876, 45012, 72216, 118524, 191178, 308346, 497559, 795318, 1276752, 2033294, 3242088, 5150502, 8166487, 12934446, 20427141, 32240122, 50764914, 79850106, 125402323, 196667538

Offset

0,3

Links

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

Index entries for linear recurrences with constant coefficients, signature (0,6,3,-15,-12,17,18,-9,-11,3,3,-1).

Formula

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

a(n) = 6*a(n-2) + 3*a(n-3) - 15*a(n-4) - 12*a(n-5) + 17*a(n-6) + 18*a(n-7) - 9*a(n-8) - 11*a(n-9) + 3*a(n-10) + 3*a(n-11) - a(n-12).

Mathematica

CoefficientList[Series[(1-x^2)*((1-x^2)^2+3*x^3)/((1-x^2)^2-x^3)^3, {x, 0, 60}], x] (* Vincenzo Librandi, Jan 07 2026 *)

Prog

(PARI) my(N=40, x='x+O('x^N)); Vec((1-x^2)*((1-x^2)^2+3*x^3)/((1-x^2)^2-x^3)^3)
(Magma) m:=60; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!(1-x^2) * ((1-x^2)^2 + 3*x^3) / ((1-x^2)^2 - x^3)^3); // Vincenzo Librandi, Jan 07 2026

Crossrefs

Cf. A391963, A392252, A392270.

Cf. A062200, A392267.

Sequence in context: A319446 A181372 A168426 * A065931 A137497 A391836

Adjacent sequences: A392266 A392267 A392268 * A392270 A392271 A392272

Keyword

nonn,easy

Author

Seiichi Manyama, Jan 05 2026

Status

approved