A389942 G.f. A(x) satisfies A(x) = 1 + x*(1-x^2)^2*A(x)^3.

1, 1, 3, 10, 43, 202, 1006, 5202, 27655, 150192, 829678, 4647342, 26334058, 150685297, 869468278, 5053346800, 29556408331, 173838629041, 1027524962368, 6100455802246, 36363248305330, 217534410727201, 1305615432843738, 7859599685354056, 47443293954331878

Offset

0,3

Links

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

Formula

a(n) = Sum_{k=0..floor(n/2)} (-1)^k * binomial(2*(n-2*k),k) * A001764(n-2*k).

D-finite with recurrence 2*n*(2*n+1)*(n-2)*a(n) -3*(n-2)*(3*n-1)*(3*n-2)*a(n-1) -2*n*(2*n+1)*(n-2)*a(n-2) +3*(27*n^3-135*n^2+184*n-24)*a(n-3) +3*(-27*n^3+189*n^2-340*n+20)*a(n-5) +3*n*(3*n-11)*(3*n-16)*a(n-7)=0. - R. J. Mathar, Oct 24 2025

Mathematica

Table[Sum[(-1)^k*Binomial[2*(n-2*k), k]*Binomial[3*(n-2*k), n-2*k]/(2*(n-2*k)+1), {k, 0, Floor[n/2]}], {n, 0, 25}] (* Vincenzo Librandi, Nov 13 2025 *)

Prog

(PARI) a(n) = sum(k=0, n\2, (-1)^k*binomial(2*(n-2*k), k)*binomial(3*(n-2*k), n-2*k)/(2*(n-2*k)+1));
(Magma) [&+[(-1)^k*Binomial(2*(n-2*k), k)*Binomial(3*(n-2*k), n-2*k)/(2*(n-2*k)+1): k in [0..Floor(n/2)]] : n in [0..30] ]; // Vincenzo Librandi, Nov 13 2025

Crossrefs

Cf. A001764, A385065.

Sequence in context: A151084 A151085 A082936 * A390170 A347006 A205487

Adjacent sequences: A389939 A389940 A389941 * A389943 A389944 A389945

Keyword

nonn,easy

Author

Seiichi Manyama, Oct 20 2025

Status

approved