A390133 G.f. A(x) satisfies A(x) = 1 + x/(1+x^2)^3 * A(x)^2.

1, 1, 2, 2, 2, 3, 6, 14, 34, 83, 204, 505, 1258, 3152, 7940, 20100, 51114, 130525, 334590, 860728, 2221432, 5750503, 14927470, 38849487, 101349738, 264987476, 694268316, 1822504484, 4792857868, 12625700121, 33312406620, 88024731565, 232923660242

Offset

0,3

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1500

Formula

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

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

Mathematica

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

Prog

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

Crossrefs

Cf. A000108, A390102.

Sequence in context: A218694 A143596 A342763 * A091712 A125721 A049798

Adjacent sequences: A390130 A390131 A390132 * A390134 A390135 A390136

Keyword

nonn

Author

Seiichi Manyama, Oct 26 2025

Status

approved