This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A384437 #33 May 29 2025 14:40:30 %S A384437 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, %T A384437 1,26,4433,627382,1733677,132,1,1,1,37,16401,18245201,4138659802, %U A384437 1816333805,429,1,1,1,50,48205,256754526,1197172898385,244829520301060,7526310334829,1430,1 %N 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. %F A384437 A(n,k) = q_binomial(2*n, n, k)/q_binomial(n+1, 1, k). %e A384437 Square array begins: %e A384437 1, 1, 1, 1, 1, 1, ... %e A384437 1, 1, 1, 1, 1, 1, ... %e A384437 1, 2, 5, 10, 17, 26, ... %e A384437 1, 5, 93, 847, 4433, 16401, ... %e A384437 1, 14, 6477, 627382, 18245201, 256754526, ... %e A384437 1, 42, 1733677, 4138659802, 1197172898385, 100333200992026, ... %o A384437 (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)))); %o A384437 (Sage) %o A384437 from sage.combinat.q_analogues import q_catalan_number %o A384437 def a(n, k): return q_catalan_number(n, k) %Y A384437 Columns k=0..12 give A000012, A000108, A015030, A015033, A015034, A015035, A015037, A015038, A015039, A015040, A015041, A015042, A015055. %Y A384437 Main diagonal gives A384282. %Y A384437 Cf. A129175 %K A384437 nonn,tabl %O A384437 0,9 %A A384437 _Seiichi Manyama_, May 29 2025