Solution of complex recurrence relation
Everything becomes much simpler if when $n$ is odd we stop the sum at $\lfloor n/2 \rfloor$, and when $k$ is even we discount the term corresponding to $k/2$ by a half. In that case you can write
$$ 2T(k) = \sum_{i=1}^{k-1} \binom{k}{i} T(i) T(k-i) $$
and so, putting $T(0) = 0$, we get that for $k \geq 2$,
$$ 2T(k) = \sum_{i=0}^k \binom{k}{i} T(i) T(k-i). \tag{1}$$
(When $k = 1$ this reads $2T(1) = 0$.)

Now define the exponential generating function
$$ P = \sum_{n \geq 0} T(n) \frac{x^n}{n!}. $$
Put $T(1) = c$. The identity (1) implies the equation
$$ 2(P-cx) = P^2. $$
Solving the quadratic, we obtain
$$ P = 1 - \sqrt{1 - 2cx} = 1 - \sum_{n \geq 0} \frac{(2n)!}{(1-2n)(n!)^24^n} (2cx)^n = \sum_{n \geq 1} \frac{(2n)!(c/2)^n}{(2n-1)n!} \frac{x^n}{n!}. $$
We conclude that for $n \geq 1$,
$$
T(n) = \frac{(2n)!(c/2)^n}{(2n-1)n!} \sim \frac{1}{\sqrt{2} n} \left( \frac{2c}{e} n \right)^n.
$$
The first few terms (starting with $T(1)$) are
$$
c,c^2,3c^3,15c^4,105c^5,\ldots
$$