%I #43 Dec 20 2023 19:05:55
%S 1,1,3,30,1001,111384,41314284,51067020290,210309203300625,
%T 2885318087540733000,131857099297936066411200,
%U 20070377346929658409924542720,10174783866874800701945612292557712,17178820188393063395267380511228827387600,96592800670609299321035523895170598736583965100
%N 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}.
%C 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.
%C [ 5, 14, 42]
%C a(3) = det( [14, 42, 132] ) = 30.
%C [42, 132, 429]
%H Alois P. Heinz, <a href="/A355400/b355400.txt">Table of n, a(n) for n = 0..62</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Lattice_path#Counting_lattice_paths">Counting lattice paths</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Hankel_matrix">Hankel matrix</a>
%F a(n) = Product_{i=1..n-1, j=i..n-1} (i+j+2*n)/(i+j).
%F a(n) mod 2 = 1 <=> n in { A131577 }.
%F 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
%e a(0) = 1: ( ).
%e a(1) = 1: (/\).
%e a(2) = 3: /\ /\ /\
%e (/\/\, /\/\), (/\/\, / \), (/ \, / \).
%e G.f. = 1 + x + 3*x^2 + 30*x^3 + 1001*x^4 + 111384*x^5 + 41314284*x^6 + ... - _Michael Somos_, Jun 27 2023
%p a:= n-> mul(mul((i+j+2*n)/(i+j), j=i..n-1), i=1..n-1):
%p seq(a(n), n=0..14);
%t 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 *)
%o (PARI) a(n) = prod(i=1, n-1, prod(j=i, n-1, (i+j+2*n)/(i+j))); \\ _Michel Marcus_, Jul 05 2022
%Y A diagonal of A078920 or of A123352 or of A368025.
%Y Cf. A000108, A006125, A074962, A076113, A110131, A112332, A131577, A358597, A368298.
%K nonn
%O 0,3
%A _Alois P. Heinz_, Jun 30 2022