Intended solution
-----------------
The Catalan numbers satisfy the reccurence
C_0 = 1 and
C_{n+1} = \sum_{k=0}^{n} C_k * C_{n-k},
which means that the coefficients of S_n are just the Catalan numbers
shifted one place to the right.

A more refined way of seeing this is to note that the generating
function for
the Catalan numbers satisfies g(s) = 1 + s*g(s)^2, which means
g(s)^2  = (g(s) - 1)/s. Now, g(s) = 1 + s + 2s^2 + 5s^3 + 14s^4 + ...,
so (g(s) - 1)/s = 1 + 2s + 5s^2 + 14s^3 + ...