A387063 G.f. A(x) satisfies A(x) = 1 + x/(1-x^2)^2 * A(x)^2.

1, 1, 2, 7, 22, 75, 264, 958, 3558, 13463, 51722, 201223, 791192, 3139131, 12552204, 50533205, 204654950, 833219825, 3408319290, 14000834736, 57732586382, 238884001905, 991558312956, 4127602773465, 17227553257504, 72078391054381, 302247212118142, 1270059248710872, 5347199223168084

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)^2)).

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

Mathematica

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

Crossrefs

Cf. A006319, A390103.

Cf. A085139, A390102.

Cf. A000108.

Sequence in context: A390198 A007141 A278151 * A090831 A174403 A384967

Adjacent sequences: A387060 A387061 A387062 * A387064 A387065 A387066

Keyword

nonn

Author

Seiichi Manyama, Oct 25 2025

Status

approved