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 A128562 #16 Jun 02 2020 01:57:40
%S A128562 1,1,1,1,2,1,1,4,4,1,1,6,12,6,1,1,10,29,29,10,1,1,14,61,94,61,14,1,1,
%T A128562 21,120,263,263,120,21,1,1,29,222,645,910,645,222,29,1,1,41,392,1468,
%U A128562 2724,2724,1468,392,41,1,1,55,669,3113,7352,9686,7352,3113,669,55,1
%N A128562 Triangle, read by rows, where T(n,k) is the coefficient of q^((n+1)*k) in the q-binomial coefficient [2*n+1, n] for n >= k >= 0.
%C A128562 Row sums equal a shifted version of A003239 (number of rooted planar trees with n non-root nodes). Column 1 is a shifted version of A000065 (-1 + number of partitions of n). Column 2 is a shifted version of A128563. This array is a variant of triangles A128545 and A047812 (Parker's partition triangle).
%F A128562 T(n,k) = [q^((n+1)*k)] Product_{j=n+1..2*n+1}(1-q^j) / Product_{j=1..n+1}(1-q^j).
%e A128562 Triangle T(n,k) (with rows n >= 0 and columns k = 0..n) begins:
%e A128562 1;
%e A128562 1, 1;
%e A128562 1, 2, 1;
%e A128562 1, 4, 4, 1;
%e A128562 1, 6, 12, 6, 1;
%e A128562 1, 10, 29, 29, 10, 1;
%e A128562 1, 14, 61, 94, 61, 14, 1;
%e A128562 1, 21, 120, 263, 263, 120, 21, 1;
%e A128562 1, 29, 222, 645, 910, 645, 222, 29, 1;
%e A128562 1, 41, 392, 1468, 2724, 2724, 1468, 392, 41, 1;
%e A128562 1, 55, 669, 3113, 7352, 9686, 7352, 3113, 669, 55, 1;
%e A128562 ...
%o A128562 (PARI) T(n,k)=if(n<k || k<0,0,if(n==0,1,polcoeff(prod(j=n+1,2*n+1,1-q^j)/prod(j=1,n+1,1-q^j),(n+1)*k,q)))
%Y A128562 Cf. A000065 (column 1), A003239 (row sums), A128563 (column 2).
%Y A128562 Variants are A047812 and A128545.
%K A128562 nonn,tabl
%O A128562 0,5
%A A128562 _Paul D. Hanna_, Mar 10 2007
%E A128562 Minor edits by _Petros Hadjicostas_, Jun 01 2020