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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}.

%I A355400 #45 Nov 16 2024 19:13:57
%S A355400 1,1,3,30,1001,111384,41314284,51067020290,210309203300625,
%T A355400 2885318087540733000,131857099297936066411200,
%U A355400 20070377346929658409924542720,10174783866874800701945612292557712,17178820188393063395267380511228827387600,96592800670609299321035523895170598736583965100
%N 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}.
%C A355400 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 A355400               [ 5,  14,  42]
%C A355400   a(3) = det( [14,  42, 132] ) = 30.
%C A355400               [42, 132, 429]
%H A355400 Alois P. Heinz, <a href="/A355400/b355400.txt">Table of n, a(n) for n = 0..62</a>
%H A355400 Wikipedia, <a href="https://en.wikipedia.org/wiki/Lattice_path#Counting_lattice_paths">Counting lattice paths</a>
%H A355400 Wikipedia, <a href="https://en.wikipedia.org/wiki/Hankel_matrix">Hankel matrix</a>
%F A355400 a(n) = Product_{i=1..n-1, j=i..n-1} (i+j+2*n)/(i+j).
%F A355400 a(n) mod 2 = 1 <=> n in { A131577 }.
%F A355400 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 A355400 a(0) = 1:  ( ).
%e A355400 a(1) = 1:  (/\).
%e A355400 a(2) = 3:                        /\      /\    /\
%e A355400            (/\/\, /\/\), (/\/\, /  \), (/  \, /  \).
%e A355400 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 A355400 a:= n-> mul(mul((i+j+2*n)/(i+j), j=i..n-1), i=1..n-1):
%p A355400 seq(a(n), n=0..14);
%t A355400 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 A355400 (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 A355400 A diagonal of A078920 or of A123352 or of A368025.
%Y A355400 Cf. A000108, A006125, A074962, A076113, A110131, A112332, A131577, A358597, A368298, A378113.
%K A355400 nonn
%O A355400 0,3
%A A355400 _Alois P. Heinz_, Jun 30 2022