A392253 a(n) = Sum_{k=0..floor(n/2)} binomial(k,3*(n-2*k)).

1, 0, 1, 0, 1, 0, 1, 1, 1, 4, 1, 10, 1, 20, 2, 35, 8, 56, 29, 84, 85, 121, 211, 175, 463, 275, 925, 506, 1718, 1079, 3017, 2457, 5097, 5565, 8464, 12121, 14197, 25142, 24753, 49725, 45697, 94334, 89150, 173166, 180254, 310974, 368734, 553455, 748924, 989759

Offset

0,10

Links

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

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

Formula

G.f.: (1-x^2)^2 / ((1-x^2)^3 - x^7).

a(n) = 3*a(n-2) - 3*a(n-4) + a(n-6) + a(n-7).

a(2*n) = A293169(n).

Mathematica

CoefficientList[Series[(1-x^2)^2/((1-x^2)^3-x^7), {x, 0, 50}], x] (* Vincenzo Librandi, Jan 06 2026 *)

Prog

(PARI) a178618(n, k) = sum(j=0, k, (-1)^(k-j)*binomial(n+1, k-j)*binomial(n+3*j, 3*j));
my(A=1, B=1, C=A^3*B, N=1, M=50, x='x+O('x^M), X=1-A*x^2, Y=7); Vec(sum(k=0, (2*N)\3, C^k*a178618(N-1, k)*X^(2*N-3*k)*x^(Y*k))/(X^3-C*x^Y)^N)
(Magma) m:=50; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!(1-x^2)^2 / ((1-x^2)^3 - x^7)); // Vincenzo Librandi, Jan 06 2026

Crossrefs

Cf. A003522, A392271, A392272, A392273.

Cf. A392254, A392255.

Cf. A178618, A293169.

Sequence in context: A059926 A306546 A138775 * A209385 A121529 A304429

Adjacent sequences: A392250 A392251 A392252 * A392254 A392255 A392256

Keyword

nonn

Author

Seiichi Manyama, Jan 04 2026

Status

approved