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
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
Programs
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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))));
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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).