A391902 a(n) = Sum_{k=0..n} binomial(3*k,3*(n-k)).

1, 1, 2, 21, 86, 305, 1381, 6042, 24901, 105364, 450017, 1902321, 8044478, 34114553, 144535714, 612041073, 2592800365, 10984222558, 46527959417, 197093207976, 834913179137, 3536745086753, 14981831303770, 63464151953133, 268838655351310, 1138818087482257

Offset

0,3

Links

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

Index entries for linear recurrences with constant coefficients, signature (3,0,22,0,3,1).

Formula

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

G.f.: ((1-x-x^2)^2 - 9*x^3) / ((1-x-x^2)^3 - 27*x^3).

a(n) = 3*a(n-1) + 22*a(n-3) + 3*a(n-5) + a(n-6).

Mathematica

CoefficientList[Series[((1-x-x^2)^2-9*x^3)/((1-x-x^2)^3-27*x^3), {x, 0, 50}], x] (* Vincenzo Librandi, Jan 01 2026 *)

Prog

(PARI) my(N=30, x='x+O('x^N)); Vec(((1-x-x^2)^2-9*x^3)/((1-x-x^2)^3-27*x^3))
(Magma) m:=50; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!((1-x-x^2)^2 - 9*x^3) / ((1-x-x^2)^3 - 27*x^3)); // Vincenzo Librandi, Jan 01 2026

Crossrefs

Cf. A391903, A391904.

Cf. A003522.

Sequence in context: A034520 A111128 A391496 * A213827 A129556 A077209

Adjacent sequences: A391899 A391900 A391901 * A391903 A391904 A391905

Keyword

nonn

Author

Seiichi Manyama, Dec 23 2025

Status

approved