A335857 -id:A335857 - cp's OEIS frontend

A368025 Array read by ascending antidiagonals: A(n,k) is the determinant 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, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 5, 1, 1, 1, 4, 14, 14, 1, 1, 1, 5, 30, 84, 42, 1, 1, 1, 6, 55, 330, 594, 132, 1, 1, 1, 7, 91, 1001, 4719, 4719, 429, 1, 1, 1, 8, 140, 2548, 26026, 81796, 40898, 1430, 1, 1, 1, 9, 204, 5712, 111384, 884884, 1643356, 379236, 4862, 1

Offset: 0

Stefano Spezia, Dec 08 2023

This array is a variant of the triangles A078920 and A123352 extended to the trivial cases (here for k=0).

			The array begins:
  1, 1, 1,   1,    1,      1,        1, ...
  1, 1, 2,   5,   14,     42,      132, ...
  1, 1, 3,  14,   84,    594,     4719, ...
  1, 1, 4,  30,  330,   4719,    81796, ...
  1, 1, 5,  55, 1001,  26026,   884884, ...
  1, 1, 6,  91, 2548, 111384,  6852768, ...
  1, 1, 7, 140, 5712, 395352, 41314284, ...
  ...

Arthur T. Benjamin, Naiomi T. Cameron, Jennifer J. Quinn, and Carl R. Yerger, Catalan determinants-a combinatorial approach, Congressus Numerantium 200, 27-34 (2010). On ResearchGate.
Jishe Feng, The explicit formula of Hankel determinant with Catalan elements, arXiv:2010.06586 [math.GM], 2020.
M. E. Mays and Jerzy Wojciechowski, A determinant property of Catalan numbers. Discrete Math. 211, No. 1-3, 125-133 (2000).
Wikipedia, Hankel matrix.

Cf. A000108 (n=1), A005700 (n=2), A006149 (n=3), A006150 (n=4), A006151 (n=5).
Cf. A000012 (k=0 or k=1 or n=0), A000330, A078920, A091962, A123352, A335857 (k=6).
Cf. A355400, A368026 (permanent), A378112.
Antidiagonal sums give A355503.

A:= proc(n, k) option remember; `if`(k=0, 1, 2^n*mul(
      (2*(k-i)+2*n-3)/(k+2*n-1-i), i=0..n-1)*A(n, k-1))
    end:
seq(seq(A(d-k, k), k=0..d), d=0..10);  # Alois P. Heinz, Dec 20 2023

A[n_,k_]:=If[n==0,1,Det[Table[CatalanNumber[i+j+k],{i,0,n-1},{j,0,n-1}]]]; Table[A[n-k,k],{n,0,11},{k,0,n}]//Flatten

For an explicit formula of A(n,k), see equation (5) in Feng, 2020.
A(n,2) = n + 1.
A(n,3) = A000330(n+1).
A(n,4) = A006858(n+1).
A(n,5) = A091962(n+1).
Diagonal: A(n,n) = A123352(2*n-1,n-1) = A355400(n).

A368024 a(n) 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+6) with i,j = 0, ..., n-1.

Original entry on oeis.org

1, 132, 372801, 18271508684, 14570336513383508, 184204867131613485842464, 36494318768452684668237864399892, 112700882376631374264115400599310944646268, 5412697889621813132124427516447652973723355158580585, 4039897382110972290799421201399595435416108353911344509968785100

Offset: 0

Stefano Spezia, Dec 08 2023

nonn

			a(4) = 14570336513383508:
   132,   429,  1430,   4862;
   429,  1430,  4862,  16796;
  1430,  4862, 16796,  58786;
  4862, 16796, 58786, 208012.

Arthur T. Benjamin, Naiomi T. Cameron, Jennifer J. Quinn, and Carl R. Yerger, Catalan determinants-a combinatorial approach, Congressus Numerantium 200, 27-34 (2010). On ResearchGate.
M. E. Mays and Jerzy Wojciechowski, A determinant property of Catalan numbers. Discrete Math. 211, No. 1-3, 125-133 (2000).
Wikipedia, Hankel matrix.

Cf. A000108, A335857 (determinant).
Cf. A278843, A368012, A368019, A368021, A368022, A368023, A368025.
Column k=6 of A368026.

Join[{1},Table[Permanent[Table[CatalanNumber[i+j+6],{i,0,n-1},{j,0,n-1}]],{n,9}]]

C(n) = binomial(2*n, n)/(n+1); \\ A000108
a(n) = matpermanent(matrix(n, n, i, j, C(i+j+4))); \\ Michel Marcus, Dec 11 2023

cp's OEIS Frontend

A368025 Array read by ascending antidiagonals: A(n,k) is the determinant 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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula