0,3

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

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

a(n) = Sum_{k=0..floor(n/3)} binomial(2*n-5*k-1,k) * Catalan(n-3*k).

D-finite with recurrence: (n - 5)*a(n) - 3*n*a(n + 3) + (-10 - 4*n)*a(n + 5) + (15 + 3*n)*a(n + 6) + (34 + 4*n)*a(n + 8) + (-10 - n)*a(9 + n) = 0. - Robert Israel, Feb 15 2026

f:= gfun:-rectoproc({(n - 5)*a(n) - 3*n*a(n + 3) + (-10 - 4*n)*a(n + 5) + (15 + 3*n)*a(n + 6) + (34 + 4*n)*a(n + 8) + (-10 - n)*a(9 + n), a(0) = 1, a(1) = 1, a(2) = 2, a(3) = 5, a(4) = 16, a(5) = 50, a(6) = 162, a(7) = 544, a(8) = 1870, a(9) = 6551}, a(n), remember):

map(f, [$0..30]); # Robert Israel, Feb 15 2026

Table[Sum[ Binomial[2*n-5*k-1, k]*CatalanNumber[n-3*k], {k, 0, Floor[n/3]}], {n, 0, 30}] (* Vincenzo Librandi, Oct 30 2025 *)

(PARI) a(n) = sum(k=0, n\3, binomial(2*n-5*k-1, k)*binomial(2*(n-3*k), n-3*k)/(n-3*k+1));

(Magma) [&+[Catalan(n-3*k) * Binomial(2*n-5*k-1, k): k in [0..Floor(n/3)]] : n in [0..30] ]; // Vincenzo Librandi, Oct 30 2025

Cf. A006319, A387063.

Cf. A000108.

Sequence in context: A148383 A148384 A389873 * A148385 A205501 A118973

Adjacent sequences: A390100 A390101 A390102 * A390104 A390105 A390106

Seiichi Manyama, Oct 25 2025

approved