A368298 -id:A368298 - cp's OEIS frontend

A368026 Array read by ascending antidiagonals: A(n, k) is the permanent of the n-th order Hankel matrix of Catalan numbers M(n) whose generic element is given by M(i,j) = A000108(i+j+k) with i,j = 0, ..., n-1.

1, 1, 1, 3, 1, 1, 95, 9, 2, 1, 38057, 979, 53, 5, 1, 207372681, 1417675, 19148, 406, 14, 1, 15977248385955, 28665184527, 97432285, 490614, 3612, 42, 1, 17828166968924572623, 8325587326635565, 7146659536022, 8755482505, 14798454, 35442, 132, 1, 292842668371666277607183121, 35389363346700690999467, 7683122105385590481, 2318987094804471, 930744290905, 499114473, 372801, 429, 1

Offset: 0

Stefano Spezia, Dec 08 2023

			The array begins:
      1,       1,        1,          1,            1, ...
      1,       1,        2,          5,           14, ...
      3,       9,       53,        406,         3612, ...
     95,     979,    19148,     490614,     14798454, ...
  38057, 1417675, 97432285, 8755482505, 930744290905, ...
  ...

Wikipedia, Hankel matrix.

Cf. A000012 (n=0), A000108 (n=1).
Cf. A368012 (k=0), A368019 (k=1), A278843 (k=2), A368021 (k=3), A368022 (k=4), A368023 (k=5), A368024 (k=6).
Cf. A368025 (determinant), A368298 (diagonal).

with(LinearAlgebra):
C:= proc(n) option remember; binomial(2*n, n)/(n+1) end:
A:= (n, k)-> `if`(n=0, 1, Permanent(Matrix(n, (i, j)-> C(i+j+k-2)))):
seq(seq(A(d-k, k), k=0..d), d=0..8);  # Alois P. Heinz, Dec 20 2023

A[n_,k_]:=If[n==0,1,Permanent[Table[CatalanNumber[i+j+k],{i,0,n-1},{j,0,n-1}]]]; Table[A[n-k,k],{n,0,8},{k,0,n}]//Flatten

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}.

Original entry on oeis.org

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

cp's OEIS Frontend

A368026 Array read by ascending antidiagonals: A(n, k) is the permanent of the n-th order Hankel matrix of Catalan numbers M(n) whose generic element is given by M(i,j) = A000108(i+j+k) with i,j = 0, ..., n-1.

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