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A358597 Number of n-tuples (p_1, p_2, ..., p_n) of Dyck paths of semilength n+1, such that each p_i is never below p_{i-1}.

%I A358597 #16 Sep 04 2023 19:37:31
%S A358597 1,2,14,330,26026,6852768,6018114036,17618122000050,
%T A358597 171879976152056250,5586863607659640852000,
%U A358597 604960371578930672694585600,218201797452928091289631307694720,262138086905421645845923269465748817136,1048861003938217198101763464819634006647101600
%N A358597 Number of n-tuples (p_1, p_2, ..., p_n) of Dyck paths of semilength n+1, such that each p_i is never below p_{i-1}.
%H A358597 Alois P. Heinz, <a href="/A358597/b358597.txt">Table of n, a(n) for n = 0..61</a>
%F A358597 a(n) = A078920(2n,n) = A123352(2n,n).
%F A358597 a(n) ~ exp(1/24) * 2^(2/3 + 5*n + 8*n^2) / (sqrt(A) * n^(1/24) * 3^(9*n^2/2 + 3*n + 5/12)), where A = A074962 is the Glaisher-Kinkelin constant. - _Vaclav Kotesovec_, Aug 26 2023
%e A358597 a(0) = 1:  ().
%e A358597                      /\
%e A358597 a(1) = 2:  (/\/\), (/  \).
%p A358597 a:= n-> mul(mul((i+j+2*n)/(i+j), j=i..n), i=1..n):
%p A358597 seq(a(n), n=0..14);
%t A358597 Join[{1}, Table[2^(3/2)*n*Gamma[4*n] * BarnesG[n+1] * Sqrt[Gamma[2*n] * BarnesG[4*n]] / BarnesG[3*n + 2], {n, 1, 12}]] (* _Vaclav Kotesovec_, Aug 26 2023 *)
%Y A358597 Cf. A074962, A078920, A123352, A355400.
%K A358597 nonn
%O A358597 0,2
%A A358597 _Alois P. Heinz_, Feb 24 2023