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}.

1, 2, 14, 330, 26026, 6852768, 6018114036, 17618122000050, 171879976152056250, 5586863607659640852000, 604960371578930672694585600, 218201797452928091289631307694720, 262138086905421645845923269465748817136, 1048861003938217198101763464819634006647101600

Offset: 0

Author

Alois P. Heinz, Feb 24 2023

Keywords

• nonn

Examples

			a(0) = 1:  ().
                     /\
a(1) = 2:  (/\/\), (/  \).
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..61

Crossrefs

Cf. A074962, A078920, A123352, A355400.

Programs

Maple

a:= n-> mul(mul((i+j+2*n)/(i+j), j=i..n), i=1..n):
seq(a(n), n=0..14);

Mathematica

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 *)

Formula

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

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