So I made an error on the question here: A closed form for $T_N = 1 + \sum\limits_{k=0}^{N-2}{(N-1-k)T_k}$?
The correct formula I'm trying to solve is more complicated and as follows:
$T_0 = T_1 = 1 $
$T_{N+1} = T_N + \sum_{k=1}^{N}T_{k-1}T_{N-k}$
I calculated the first few terms:
$T_2 = 2$
$T_3 = 2 + T_0T_1 + T_1T_0 = 4$
$T_4 = 4 + T_0T_2 + T_1T_1 + T_0T_2 = 4 + 2 + 1 + 2 = 9$
$T_5 = 9 + T_0T_3 + T_1T_2 + T_2T_1 + T_3T_0 = 9 + 4 + 2 + 2 + 4 = 21$
and plugged it into OEIS and it gave me back: http://oeis.org/A001006 . Which is exactly the problem I am trying to solve (non-intersecting chords on a circle).
What I don't understand is how to use the FORMULA section from OEIS. Would someone be kind enough to take one of the formulas from OEIS and walk me through how to use it to actually close my recurrence, so I can efficiently calculate $T_N$ for large N.