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%I #27 Apr 02 2021 03:35:55
%S 1,2,5,22,177,2606,70226,3457742,311348897,51177188350,15377065068510,
%T 8430169458379450,8446194335222422950,15435904380166258833482,
%U 51546769958534244310727102,313937270864810066000897492222,3493348088919874482660174997662017
%N a(n) = 1 + Sum_{j=1..n} Product_{k=0..j-1} binomial(2*n-1,n+k) / binomial(2*n-1,k).
%H Seiichi Manyama, <a href="/A342967/b342967.txt">Table of n, a(n) for n = 0..87</a>
%F a(n) = Sum_{j=0..n} Product_{k=0..n-1} binomial(n+k,j)/binomial(j+k,j).
%F a(n) ~ c * exp(1/12) * 2^(4*n^2 - 1/12) / (A * n^(1/12) * 3^(9*n^2/4 - 1/6)), where c = JacobiTheta3(0,1/3) = EllipticTheta[3, 0, 1/3] = 1.69145968168171534134842... if n is even, and c = JacobiTheta2(0,1/3) = EllipticTheta[2, 0, 1/3] = 1.69061120307521423305296... if n is odd, and A is the Glaisher-Kinkelin constant A074962. - _Vaclav Kotesovec_, Apr 02 2021
%t a[n_] := 1 + Sum[Product[Binomial[2*n - 1, n + k]/Binomial[2*n - 1, k], {k, 0, j - 1}], {j, 1, n}]; Array[a, 17, 0] (* _Amiram Eldar_, Apr 01 2021 *)
%t Table[1 + BarnesG[2*n + 1] * Sum[BarnesG[j + 1]*BarnesG[n - j + 1] / (BarnesG[n + j + 1]*BarnesG[2*n - j + 1]), {j, 1, n}], {n, 0, 15}] (* _Vaclav Kotesovec_, Apr 02 2021 *)
%o (PARI) a(n) = 1+sum(j=1, n, prod(k=0, j-1, binomial(2*n-1, n+k)/binomial(2*n-1, k)));
%o (PARI) a(n) = sum(j=0, n, prod(k=0, n-1, binomial(n+k, j)/binomial(j+k, j)));
%Y Row sums of A342972.
%Y Cf. A001181, A005362, A005363, A005364, A005365, A005366, A116925.
%K nonn
%O 0,2
%A _Seiichi Manyama_, Apr 01 2021
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