A392043 a(n) = Sum_{k=0..floor(n/3)} (k+1) * 2^k * 3^(n-3*k) * binomial(k,3*(n-3*k)).

1, 0, 0, 4, 0, 0, 12, 0, 0, 32, 96, 0, 80, 960, 0, 192, 5760, 0, 448, 26880, 4032, 1024, 107520, 64512, 2304, 387072, 580608, 5120, 1290240, 3870720, 149504, 4055040, 21288960, 3065856, 12165120, 102187008, 36548608, 35143680, 442810368, 316407808, 102715392

Offset

0,4

Links

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

Index entries for linear recurrences with constant coefficients, signature (0,0,12,0,0,-60,0,0,160,48,0,-240,-288,0,192,576,0,-64,-384,-576).

Formula

G.f.: (1-2*x^3) * ((1-2*x^3)^3 + 48*x^10) / ((1-2*x^3)^3 - 24*x^10)^2.

a(n) = 12*a(n-3) - 60*a(n-6) + 160*a(n-9) + 48*a(n-10) - 240*a(n-12) - 288*a(n-13) + 192*a(n-15) + 576*a(n-16) - 64*a(n-18) - 384*a(n-19) - 576*a(n-20).

Mathematica

CoefficientList[Series[(1-2*x^3)*((1-2*x^3)^3+48*x^10)/((1-2*x^3)^3-24*x^10)^2, {x, 0, 50}], x] (* Vincenzo Librandi, Dec 31 2025 *)

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=2, B=3, C=A^3*B, N=2, M=50, x='x+O('x^M), X=1-A*x^3, Y=10); 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!(1-2*x^3) * ((1-2*x^3)^3 + 48*x^10) / ((1-2*x^3)^3 - 24*x^10)^2); // Vincenzo Librandi, Dec 31 2025

Crossrefs

Cf. A178618, A392042.

Sequence in context: A035539 A178517 A392075 * A049207 A092219 A391960

Adjacent sequences: A392040 A392041 A392042 * A392044 A392045 A392046

Keyword

nonn

Author

Seiichi Manyama, Dec 29 2025

Status

approved