A355400 Number of n-tuples (p_1, p_2, ..., p_n) of Dyck paths of semilength n, such that each p_i is never below p_{i-1}.

1, 1, 3, 30, 1001, 111384, 41314284, 51067020290, 210309203300625, 2885318087540733000, 131857099297936066411200, 20070377346929658409924542720, 10174783866874800701945612292557712, 17178820188393063395267380511228827387600, 96592800670609299321035523895170598736583965100

Offset: 0

Author

Alois P. Heinz, Jun 30 2022

Keywords

• nonn

Comments

Determinant of the n X n Hankel matrix whose i-th antidiagonal is filled with the n+i-th Catalan number for i = 0..2*n-2.

[ 5, 14, 42]

a(3) = det( [14, 42, 132] ) = 30.

[42, 132, 429]

Examples

			a(0) = 1:  ( ).
a(1) = 1:  (/\).
a(2) = 3:                        /\      /\    /\
           (/\/\, /\/\), (/\/\, /  \), (/  \, /  \).
G.f. = 1 + x + 3*x^2 + 30*x^3 + 1001*x^4 + 111384*x^5 + 41314284*x^6 + ... - _Michael Somos_, Jun 27 2023
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..62
• Wikipedia, Counting lattice paths
• Wikipedia, Hankel matrix

Crossrefs

A diagonal of A078920 or of A123352 or of A368025.

Cf. A000108, A006125, A074962, A076113, A110131, A112332, A131577, A358597, A368298, A378113.

Programs

Maple

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

Mathematica

Join[{1}, Table[Sqrt[2*BarnesG[4*n]] * BarnesG[n] * Gamma[2*n]^(3/2) / BarnesG[3*n + 1], {n, 1, 12}]] (* Vaclav Kotesovec, Aug 26 2023 *)

PARI

a(n) = prod(i=1, n-1, prod(j=i, n-1, (i+j+2*n)/(i+j))); \\ Michel Marcus, Jul 05 2022

Formula

a(n) = Product_{i=1..n-1, j=i..n-1} (i+j+2*n)/(i+j).

a(n) mod 2 = 1 <=> n in { A131577 }.

a(n) ~ exp(1/24) * 2^(1/6 - n + 8*n^2) / (sqrt(A) * n^(1/24) * 3^(9*n^2/2 - 1/12)), where A = A074962 is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Aug 26 2023