Consider the following recurrence relation defined for two integer variables $H,n \geq 0$: \begin{equation} \gamma(H,n) = \sum_{K=0}^{\lfloor H/2 \rfloor} \gamma(K,n-1) \gamma(H-K,n-1) \end{equation} with the boundary conditions $\gamma(0,n) = 1$ for every $n \geq 0$, and $\gamma(1,0) = 1$, $\gamma(H,0) = 0$ for every $H \geq 2$. I want to understand the solution to this recurrence as much as possible. Here are some easy observations.
This is a table of the first values of $\gamma(H,n)$: $$\begin{array}{r|rrrrrrrrr} H & 0& 1& 2& 3& 4& 5& 6& 7& 8\\ \hline n=0& 1& 1& 0& 0& 0& 0& 0& 0& 0\\ n=1& 1& 1& 1& 0& 0& 0& 0& 0& 0\\ n=2& 1& 1& 2& 1& 1& 0& 0& 0& 0\\ n=3& 1& 1& 3& 3& 6& 3& 3& 1& 1\\ n=4& 1& 1& 4& 6& 18& 18& 33& 31& 56\\ n=5& 1& 1& 5& 10& 40& 60& 159& 244& 651\\ \end{array}$$ By looking at this table, the recurrence relation for $\gamma(H,n)$ can be interpreted as a sum of symmetric products of pairs of numbers in the previous row $\gamma(\cdot,n-1)$ from $0$ column up to the given column $H$.
For a fixed $H$ it is easy to see that $\gamma(H,n)$ is a polynomial in $n$ and its degree is $H-1$. For example, the first polynomials read as: \begin{align} \gamma(1,n) &= 1 \\ \gamma(2,n) &= n \\ \gamma(3,n) &= \frac{1}{2}(n-1)n \\ \gamma(4,n) &= \frac{1}{2}(n-1)^2 n \\ \gamma(5,n) &= \frac{1}{4}(n-2)(n-1)^2 n \\ \gamma(6,n) &= \frac{1}{40}(n-2)(n-1) n (8n^2-21n+11) \\ \gamma(7,n) &= \frac{1}{120}(n-2)(n-1) n (14n^3-69n^2+100n-37) \\ ... \end{align}
It is also easy to see that $\gamma(H,n) = \gamma(2^n-H,n)$ for every $2^n \geq H \geq 0$ and $\gamma(H,n) = 0$ for every $H > 2^n$. Moreover, for fixed $H$ polynomials $\gamma(H,n)$ always have zeros at every $n < \log_2(H)$.
Finally, I can get an easy lower bound $\gamma(2^{n-1},n) \geq 2^{2^{n-2}}$ for every $n \geq 2$. However, I am interested in the exact asymptotics of $\gamma(H(n),n)$ for different $H(n)$ as a function of $n$.
So here is the first question. What are the asymptotics of $\gamma(H(n),n)$ in the following 3 cases: \begin{align} (1) \quad H(n) &= 2^{n-1}, \\ (2) \quad H(n) &= O(a^{n}) \text{ for some $a > 1$}, \\ (3) \quad H(n) &= O(n^d) \text{ for some $d > 0$},\\ \end{align}
A second related question is what can we say about the coefficients of the polynomial $\gamma(H,n)$ for fixed $H$? Does there exist a closed formula for them (as a function of $H$)?
