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.
0.78270418017152170184470749773460905502131295094863751477583185208650897389...
T(2,1) = 2: 12, 21 (the two U's of UDU overlap). T(3,0) = 3: 132, 213, 321. T(3,1) = 3: 123, 231, 312. T(4,0) = 12: 1243, 1342, 1432, 2134, 2143, 2431, 3124, 3214, 3421, 4213, 4312, 4321. T(4,1) = 4: 1234, 2341, 3412, 4123. T(4,2) = 8: 1324, 1423, 2314, 2413, 3142, 3241, 4132, 4231. Triangle T(n,k) begins: : 0 : 1; : 1 : 1; : 2 : 0, 2; : 3 : 3, 3; : 4 : 12, 4, 8; : 5 : 35, 45, 40; : 6 : 144, 348, 132, 96; : 7 : 910, 1862, 1316, 952; : 8 : 5976, 11600, 14808, 5760, 2176; : 9 : 39942, 100260, 123606, 63360, 35712; : 10 : 306570, 919270, 1069910, 910650, 343040, 79360;
b:= proc(u, o, t) option remember; `if`(u+o=0,
`if`(t=2, x, 1), expand(
add(b(u+j-1, o-j, 2)*`if`(t=3, x, 1), j=1..o)+
add(b(u-j, o+j-1, `if`(t=2, 3, 1)), j=1..u)))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))
(`if`(n<2, 1, n* b(0, n-1, 1))):
seq(T(n), n=0..12);
b[u_, o_, t_] := b[u, o, t] = If[u + o == 0, If[t == 2, x, 1], Expand[Sum[ b[u + j - 1, o - j, 2]*If[t == 3, x, 1], {j, 1, o}] + Sum[b[u - j, o + j - 1, If[t == 2, 3, 1]], {j, 1, u}]]];
T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]] [If[n < 2, 1, n*b[0, n - 1, 1]]];
T /@ Range[0, 12] // Flatten (* Jean-François Alcover, Dec 19 2020, after Alois P. Heinz *)