A391986 a(n) = Sum_{k=0..floor(n/4)} 2^k * 3^(n-3*k) * binomial(k+2,2) * binomial(n-2*k,k) * binomial(n-3*k,k).

1, 3, 9, 27, 117, 567, 2673, 11907, 51597, 222831, 968841, 4236219, 18551781, 81154791, 354374433, 1545091443, 6729252957, 29281369023, 127304349273, 552981937419, 2399866971381, 10405820054007, 45080675201457, 195138858704643, 844014816147501, 3647723547387663, 15753283530001065

Offset

0,2

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Formula

G.f.: ((1-3*x)^4 - 24*x^4*(1-3*x)^2 + 216*x^8) / ((1-3*x)^2 - 24*x^4)^(5/2).

Mathematica

CoefficientList[Series[((1-3*x)^4-24*x^4*(1-3*x)^2+216*x^8)/((1-3*x)^2-24*x^4)^(5/2), {x, 0, 50}], x] (* Vincenzo Librandi, Dec 31 2025 *)

Prog

(PARI) a098473(n, k) = binomial(n, k)*binomial(2*k, k);
my(A=2, B=3, C=A*B, N=2, M=30, x='x+O('x^M), X=1-B*x, Y=4); Vec(sum(k=0, N, (-C)^k*a098473(N, k)*X^(2*N-2*k)*x^(Y*k))/(X^2-4*C*x^Y)^(N+1/2))
(Magma) m := 50; R<x> := PowerSeriesRing(RationalField(), m); Coefficients(((1-3*x)^4 - 24*x^4*(1-3*x)^2 + 216*x^8) / ((1-3*x)^2 - 24*x^4)^(5/2)); // Vincenzo Librandi, Dec 31 2025

Crossrefs

Cf. A390830, A391985.

Cf. A098473, A098482.

Sequence in context: A391985 A230951 A121746 * A323927 A146151 A306681

Adjacent sequences: A391983 A391984 A391985 * A391987 A391988 A391989

Keyword

nonn

Author

Seiichi Manyama, Dec 26 2025

Status

approved