A392252 a(n) = Sum_{k=0..floor(n/2)} binomial(k+2,2) * binomial(k,2*(n-2*k)).

1, 0, 3, 0, 6, 6, 10, 30, 15, 90, 36, 210, 133, 420, 456, 784, 1305, 1512, 3205, 3240, 7041, 7590, 14433, 18150, 28801, 42108, 58020, 93132, 120240, 197106, 255482, 404670, 547518, 819160, 1164711, 1657008, 2438160, 3372630, 5017080, 6907038, 10186512, 14171910, 20521735, 28987470

Offset

0,3

Links

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

Index entries for linear recurrences with constant coefficients, signature (0,6,0,-15,3,20,-12,-15,18,3,-12,5,3,-3,1).

Formula

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

a(n) = 6*a(n-2) - 15*a(n-4) + 3*a(n-5) + 20*a(n-6) - 12*a(n-7) - 15*a(n-8) + 18*a(n-9) + 3*a(n-10) - 12*a(n-11) + 5*a(n-12) + 3*a(n-13) - 3*a(n-14) + a(n-15).

Mathematica

CoefficientList[Series[(1-x^2)*((1-x^2)^2+3*x^5)/((1-x^2)^2-x^5)^3, {x, 0, 50}], x] (* Vincenzo Librandi, Jan 06 2026 *)

Prog

(PARI) my(A=1, B=1, C=A^2*B, N=3, M=50, x='x+O('x^M), X=1-A*x^2, Y=5); 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:=50; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!(1-x^2) * ((1-x^2)^2 + 3*x^5) / ((1-x^2)^2 - x^5)^3); // Vincenzo Librandi, Jan 06 2026

Crossrefs

Cf. A391963, A392269, A392270.

Cf. A392250, A392251.

Sequence in context: A019808 A182145 A201569 * A056459 A021330 A199055

Adjacent sequences: A392249 A392250 A392251 * A392253 A392254 A392255

Keyword

nonn

Author

Seiichi Manyama, Jan 04 2026

Status

approved