A392075 a(n) = Sum_{k=0..floor(n/3)} (k+1) * 2^k * 3^(n-3*k) * binomial(k,4*(n-3*k)).

1, 0, 0, 4, 0, 0, 12, 0, 0, 32, 0, 0, 80, 240, 0, 192, 2880, 0, 448, 20160, 0, 1024, 107520, 0, 2304, 483840, 20736, 5120, 1935360, 414720, 11264, 7096320, 4561920, 24576, 24330240, 36495360, 53248, 79073280, 237219840, 1552384, 246005760, 1328431104, 40501248

Offset

0,4

Links

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

Index entries for linear recurrences with constant coefficients, signature (0,0,16,0,0,-112,0,0,448,0,0,-1120,96,0,1792,-768,0,-1792,2304,0,1024,-3072,0,-256,1536,-2304).

Formula

G.f.: (1-2*x^3)^2 * ((1-2*x^3)^4 + 144*x^13) / ((1-2*x^3)^4 - 48*x^13)^2.

a(n) = 16*a(n-3) - 112*a(n-6) + 448*a(n-9) - 1120*a(n-12) + 96*a(n-13) + 1792*a(n-15) - 768*a(n-16) - 1792*a(n-18) + 2304*a(n-19) + 1024*a(n-21) - 3072*a(n-22) - 256*a(n-24) + 1536*a(n-25) - 2304*a(n-26).

Mathematica

CoefficientList[Series[(1-2*x^3)^2*((1-2*x^3)^4+144*x^13)/((1-2*x^3)^4-48*x^13)^2, {x, 0, 50}], x] (* Vincenzo Librandi, Dec 31 2025 *)

Prog

(PARI) a178619(n, k) = sum(j=0, k, (-1)^(k-j)*binomial(n+1, k-j)*binomial(n+4*j, 4*j));
my(A=2, B=3, C=A^4*B, N=2, M=50, x='x+O('x^M), X=1-A*x^3, Y=13); Vec(sum(k=0, (3*N)\4, C^k*a178619(N-1, k)*X^(3*N-4*k)*x^(Y*k))/(X^4-C*x^Y)^N)
(Magma) m:=50; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!(1-2*x^3)^2 * ((1-2*x^3)^4 + 144*x^13) / ((1-2*x^3)^4 - 48*x^13)^2); // Vincenzo Librandi, Dec 31 2025

Crossrefs

Cf. A178619, A392043.

Sequence in context: A249346 A035539 A178517 * A392043 A049207 A092219

Adjacent sequences: A392072 A392073 A392074 * A392076 A392077 A392078

Keyword

nonn

Author

Seiichi Manyama, Dec 30 2025

Status

approved