We are given the following equation for $0 \le d \le n$ ($d,n$ integers):
\begin{align} F(n,d) & = \max_{0\le k\le n/2}(F(k,d)+F(n-2k, d-1)). \\[8pt] F(0,d) & = 1. \\[8pt] F(n,0) & = 1. \end{align}
The task is to upper bound the $F(n,d)$. The hypothesis is that $F(n,d) = O(n^\gamma \operatorname{poly}(d))$ where $\gamma$ is independent constant s.t. $0<\gamma < 1$. I'm looking for any tool for bounding/solving these type of equations.
Remarks:
1) Pessimistically, if the hypothesis is not true, we would have $F(n,d) = \Omega(n)$ or rather $F(n,d) =\Omega(n^{1-o(1)})$
2) If we change $d-1$ to $d$ in the equation the result is $\Omega(n)$ – thus, this is the crucial part of the equation.