A391960 a(n) = Sum_{k=0..floor(n/3)} (k+1) * 2^k * 3^(n-3*k) * binomial(k,2*(n-3*k)).

1, 0, 0, 4, 0, 0, 12, 36, 0, 32, 288, 0, 80, 1440, 720, 192, 5760, 8640, 448, 20160, 60480, 13120, 64512, 322560, 195840, 193536, 1451520, 1746944, 739584, 5806080, 11623424, 5253120, 21288960, 63891456, 45112320, 75727872, 306614272, 339001344, 302911488, 1328545792

Offset

0,4

Links

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

Index entries for linear recurrences with constant coefficients, signature (0,0,8,0,0,-24,24,0,32,-96,0,-16,96,-144).

Formula

G.f.: ((1-2*x^3)^2 + 12*x^7) / ((1-2*x^3)^2 - 12*x^7)^2.

a(n) = 8*a(n-3) - 24*a(n-6) + 24*a(n-7) + 32*a(n-9) - 96*a(n-10) - 16*a(n-12) + 96*a(n-13) - 144*a(n-14).

Mathematica

CoefficientList[Series[((1-2*x^3)^2+12*x^7)/((1-2*x^3)^2-12*x^7)^2, {x, 0, 50}], x] (* Vincenzo Librandi, Dec 29 2025 *)

Prog

(PARI) my(A=2, B=3, C=A^2*B, N=2, M=40, x='x+O('x^M), X=1-A*x^3, Y=7); 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!((1-2*x^3)^2 + 12*x^7) / ((1-2*x^3)^2 - 12*x^7)^2); // Vincenzo Librandi, Dec 29 2025

Crossrefs

Cf. A390775, A391961.

Cf. A390781.

Sequence in context: A392043 A049207 A092219 * A262227 A336731 A069026

Adjacent sequences: A391957 A391958 A391959 * A391961 A391962 A391963

Keyword

nonn,easy

Author

Seiichi Manyama, Dec 23 2025

Status

approved