A378113 -id:A378113 - cp's OEIS frontend

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

Alois P. Heinz, Jun 30 2022

nonn

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]

			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

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

A diagonal of A078920 or of A123352 or of A368025.

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

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

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 *)

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

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

A378112 Number A(n,k) of k-tuples (p_1, p_2, ..., p_k) of Dyck paths of semilength n, such that each p_i is never below p_{i-1} and the upper path p_k only touches the x-axis at its endpoints; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 2, 2, 0, 1, 1, 3, 9, 5, 0, 1, 1, 4, 23, 55, 14, 0, 1, 1, 5, 46, 265, 400, 42, 0, 1, 1, 6, 80, 880, 3942, 3266, 132, 0, 1, 1, 7, 127, 2347, 23695, 70395, 28999, 429, 0, 1, 1, 8, 189, 5403, 105554, 824229, 1445700, 274537, 1430, 0

Offset: 0

Alois P. Heinz, Nov 16 2024

			A(3,2) = 9:
                             /\
           /\/\             /  \      /\     /\/\
  (/\/\/\,/    \)  (/\/\/\,/    \)  (/  \/\,/    \)
.
            /\                                /\
    /\     /  \        /\   /\/\        /\   /  \
  (/  \/\,/    \)  (/\/  \,/    \)  (/\/  \,/    \)
.
                             /\        /\     /\
    /\/\   /\/\      /\/\   /  \      /  \   /  \
  (/    \,/    \)  (/    \,/    \)  (/    \,/    \)
.
Square array A(n,k) begins:
  1,  1,    1,     1,      1,       1,        1, ...
  1,  1,    1,     1,      1,       1,        1, ...
  0,  1,    2,     3,      4,       5,        6, ...
  0,  2,    9,    23,     46,      80,      127, ...
  0,  5,   55,   265,    880,    2347,     5403, ...
  0, 14,  400,  3942,  23695,  105554,   382508, ...
  0, 42, 3266, 70395, 824229, 6601728, 40446551, ...

Alois P. Heinz, Antidiagonals n = 0..115, flattened
Wikipedia, Counting lattice paths

Columns k=0-3 give: A019590(n+1), A120588, A355281, A378114.
Rows n=0+1,2,3 give: A000012, A001477, A101986.
Main diagonal gives A378113.
Cf. A000108, A078920, A123352, A368025.

b:= proc(n, k) option remember; `if`(n=0, 1, 2^k*mul(
      (2*(n-i)+2*k-3)/(n+2*k-1-i), i=0..k-1)*b(n-1, k))
    end:
A:= proc(n, k) option remember;
      b(n, k)-add(A(n-i, k)*b(i, k), i=1..n-1)
    end:
seq(seq(A(n, d-n), n=0..d), d=0..10);

Column k is INVERTi transform of row k of A368025.

cp's OEIS Frontend

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}.

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