Hi John,
I would write the code generating this sequence like so:
ClearAll["Global`*"]
b[0, _] = 1;
b[1, 0] = 0;
b[1, k_] := If[k > 0, 1, 0];
b[n_, k_] := b[n, k] = Sum[Binomial[n - 1, r - 1]*b[r - 1, k - 1]*b[n - r, k - 1], {r, 1, n}]
t[n_, k_] := b[n, k] - b[n, k - 1]
sequence = Flatten@Table[t[n, k], {n, 0, 10}, {k, 0, n}]
which - I think - is much easier to read (even though there surely might be simpler ways!). As a brute force method for getting a closed form FindSequenceFunction
can be helpful - but unfortunately not here! Maybe one gets a little "understanding" on what is going on with this:
logData = Flatten[Table[{n, k, Log[1. + t[n, k]]}, {n, 0, 10}, {k, 0, n}], 1];
route = Flatten[Table[{n, k, 14}, {n, 0, 10}, {k, 0, n}], 1];
Show[ListPlot3D[logData, ImageSize -> Large], Graphics3D[{Thick, Line[route]}]]
which gives:
Hope this helps a little, regards -- Henrik