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.
Array begins: 0,0,0,0,0,0,..., 0,1,3,7,15,31,..., 0,3,13,45,145,453,..., 0,7,45,229,1065,4717,..., 0,15,145,1065,6901,41505,..., 0,31,453,4717,41505,32946,..., ...
T[n_, k_] := Sum[j!^2*StirlingS2[k + 1, j + 1]*StirlingS2[n + 1, j + 1], {j, 1, Min[k, n]}]; Table[T[n - k, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Feb 24 2018 *)
Array begins: 0,0,0,0,0,0,..., 0,0,0,0,0,0,..., 0,0,2,12,50,180,..., 0,0,12,108,660,3420,..., 0,0,50,660,5714,40860,..., 0,0,180,3420,40860,39150,..., ...
T[n_, k_] := Sum[(1/2)*(j - 1 )*j!^2*StirlingS2[k + 1, j + 1]*StirlingS2[n + 1, j + 1], {j, 2, Min[k, n]}]; Table[T[n - k, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Feb 24 2018 *)
a[n_] := Sum[Binomial[j - 1 , k - 1 ]*j!^2*StirlingS2[n + 1, j + 1]^2/k!, {k, 0, n}, {j, k, n}]; Table[a[n], {n, 0, 15}] (* Jean-François Alcover, Feb 24 2018 *)