A391903 a(n) = Sum_{k=0..floor(n/2)} binomial(3*k,3*(n-2*k)).

1, 0, 1, 1, 1, 20, 2, 84, 85, 221, 925, 675, 5007, 5821, 19020, 49951, 72829, 296770, 428528, 1364693, 3001497, 5998390, 18112232, 31698490, 92287620, 193987770, 443261263, 1145617890, 2292692419, 6134650552, 13029745624, 31155756788, 74677089095, 162011895318, 407612202760

Offset

0,6

Links

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

Index entries for linear recurrences with constant coefficients, signature (0,3,3,-3,21,-2,3,3,1).

Formula

G.f.: ((1-x^2-x^3)^2 - 9*x^5) / ((1-x^2-x^3)^3 - 27*x^5).

a(n) = 3*a(n-2) + 3*a(n-3) - 3*a(n-4) + 21*a(n-5) - 2*a(n-6) + 3*a(n-7) + 3*a(n-8) + a(n-9).

Mathematica

CoefficientList[Series[((1-x^2-x^3)^2-9*x^5)/((1-x^2-x^3)^3-27*x^5), {x, 0, 50}], x] (* Vincenzo Librandi, Dec 30 2025 *)

Prog

(PARI) my(N=40, x='x+O('x^N)); Vec(((1-x^2-x^3)^2-9*x^5)/((1-x^2-x^3)^3-27*x^5))
(Magma) m:=50; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!((1-x^2-x^3)^2 - 9*x^5) / ((1-x^2-x^3)^3 - 27*x^5)); // Vincenzo Librandi, Dec 30 2025

Crossrefs

Cf. A391902, A391904.

Sequence in context: A303849 A277981 A040399 * A040390 A040391 A255860

Adjacent sequences: A391900 A391901 A391902 * A391904 A391905 A391906

Keyword

nonn

Author

Seiichi Manyama, Dec 23 2025

Status

approved