A359108 a(n) = A128899(2*n, n) = 2*binomial(4*n - 1, 3*n) for n >= 1 and a(0) = 1.

1, 2, 14, 110, 910, 7752, 67298, 592020, 5259150, 47071640, 423830264, 3834669566, 34834267234, 317506779800, 2902365981900, 26597044596360, 244263468539790, 2247575790712824, 20716044882791720, 191230475831922200, 1767658071106087160, 16359617358545329440

Offset: 0

Author

Peter Luschny, Dec 27 2022

Keywords

• nonn
• easy

Links

• Paolo Xausa, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A005810, A128899, A224274.

Programs

Maple

a := n -> ifelse(n = 0, 1, 2*binomial(4*n - 1, 3*n)):
# Alternative:
a := proc(n) option remember; ifelse(n < 2, n + 1, (8*(2*n - 1) * (4*n - 3) * (4*n - 1) * a(n - 1)) / (3 * n * (3*n - 2) * (3*n - 1))) end:
seq(a(n), n = 0..19);

Mathematica

A359108[n_] := If[n == 0, 1, 2*Binomial[4*n - 1, 3*n]];
Array[A359108, 25, 0] (* Paolo Xausa, Sep 18 2024 *)

Formula

a(n) = (8*(2*n - 1) * (4*n - 3) * (4*n - 1) * a(n - 1)) / (3*n * (3*n - 2) * (3*n - 1)) for n >= 2.

a(n) = (1/2)*A005810(n) = 2*A224274(n) for n >= 1. - Peter Bala, Feb 08 2023

a(n) = [x^n] C(x)^(2*n), where C(x) = (1 - sqrt(1 - 4*x))/(2*x) is the g.f. of the Catalan numbers A000108. - Peter Bala, Apr 27 2023