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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}.
4

%I #16 Sep 04 2023 19:37:31

%S 1,2,14,330,26026,6852768,6018114036,17618122000050,

%T 171879976152056250,5586863607659640852000,

%U 604960371578930672694585600,218201797452928091289631307694720,262138086905421645845923269465748817136,1048861003938217198101763464819634006647101600

%N 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 Alois P. Heinz, <a href="/A358597/b358597.txt">Table of n, a(n) for n = 0..61</a>

%F a(n) = A078920(2n,n) = A123352(2n,n).

%F 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 a(0) = 1: ().

%e /\

%e a(1) = 2: (/\/\), (/ \).

%p a:= n-> mul(mul((i+j+2*n)/(i+j), j=i..n), i=1..n):

%p seq(a(n), n=0..14);

%t 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 Cf. A074962, A078920, A123352, A355400.

%K nonn

%O 0,2

%A _Alois P. Heinz_, Feb 24 2023