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

%I A359108 #31 Mar 24 2025 05:54:43
%S A359108 1,2,14,110,910,7752,67298,592020,5259150,47071640,423830264,
%T A359108 3834669566,34834267234,317506779800,2902365981900,26597044596360,
%U A359108 244263468539790,2247575790712824,20716044882791720,191230475831922200,1767658071106087160,16359617358545329440
%N A359108 a(n) = A128899(2*n, n) = 2*binomial(4*n - 1, 3*n) for n >= 1 and a(0) = 1.
%H A359108 Paolo Xausa, <a href="/A359108/b359108.txt">Table of n, a(n) for n = 0..1000</a>
%F A359108 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.
%F A359108 a(n) = (1/2)*A005810(n) = 2*A224274(n) for n >= 1. - _Peter Bala_, Feb 08 2023
%F A359108 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
%p A359108 a := n -> ifelse(n = 0, 1, 2*binomial(4*n - 1, 3*n)):
%p A359108 # Alternative:
%p A359108 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:
%p A359108 seq(a(n), n = 0..19);
%t A359108 A359108[n_] := If[n == 0, 1, 2*Binomial[4*n - 1, 3*n]];
%t A359108 Array[A359108, 25, 0] (* _Paolo Xausa_, Sep 18 2024 *)
%Y A359108 Cf. A005810, A128899, A224274.
%K A359108 nonn,easy
%O A359108 0,2
%A A359108 _Peter Luschny_, Dec 27 2022