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A358597
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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}.
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4
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1, 2, 14, 330, 26026, 6852768, 6018114036, 17618122000050, 171879976152056250, 5586863607659640852000, 604960371578930672694585600, 218201797452928091289631307694720, 262138086905421645845923269465748817136, 1048861003938217198101763464819634006647101600
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OFFSET
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0,2
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LINKS
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FORMULA
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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
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EXAMPLE
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a(0) = 1: ().
/\
a(1) = 2: (/\/\), (/ \).
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MAPLE
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a:= n-> mul(mul((i+j+2*n)/(i+j), j=i..n), i=1..n):
seq(a(n), n=0..14);
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MATHEMATICA
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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 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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