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

0,9

Comments

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

Links

Table of n, a(n) for n=0..65.

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.

Formula

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).

Example

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, ...
  ...

Maple

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

Mathematica

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

Crossrefs

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.

Sequence in context: A305962 A144150 A124560 * A290759 A306245 A275043

Adjacent sequences: A368022 A368023 A368024 * A368026 A368027 A368028

Keyword

nonn,tabl

Author

Stefano Spezia, Dec 08 2023

Status

approved