A384437 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where A(n,k) is the n-th q-Catalan number for q=k.

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 5, 5, 1, 1, 1, 10, 93, 14, 1, 1, 1, 17, 847, 6477, 42, 1, 1, 1, 26, 4433, 627382, 1733677, 132, 1, 1, 1, 37, 16401, 18245201, 4138659802, 1816333805, 429, 1, 1, 1, 50, 48205, 256754526, 1197172898385, 244829520301060, 7526310334829, 1430, 1

Offset: 0

Author

Seiichi Manyama, May 29 2025

Keywords

• nonn
• tabl

Examples

			Square array begins:
  1,  1,       1,          1,             1,               1, ...
  1,  1,       1,          1,             1,               1, ...
  1,  2,       5,         10,            17,              26, ...
  1,  5,      93,        847,          4433,           16401, ...
  1, 14,    6477,     627382,      18245201,       256754526, ...
  1, 42, 1733677, 4138659802, 1197172898385, 100333200992026, ...
		

Crossrefs

Columns k=0..12 give A000012, A000108, A015030, A015033, A015034, A015035, A015037, A015038, A015039, A015040, A015041, A015042, A015055.

Main diagonal gives A384282.

Cf. A129175

Programs

PARI

a(n, k) = if(k==1, binomial(2*n, n)/(n+1), (1-k)/(1-k^(n+1))*prod(j=0, n-1, (1-k^(2*n-j))/(1-k^(j+1))));

Sage

from sage.combinat.q_analogues import q_catalan_number
def a(n, k): return q_catalan_number(n, k)

Formula

A(n,k) = q_binomial(2*n, n, k)/q_binomial(n+1, 1, k).