I'm trying to solve an almost-simple recurrence relation: $$ F(d,h) = F(d-1,h) + 2[d \le \alpha h] F(d-1,h-1) $$ for $h \ge 2$, $d \ge 0$, provided that $$ F(d,2) = \begin{cases} d+1, &\text{ if } d \le 2\alpha \\ 2\alpha + 1, &\text{ if } d > 2\alpha, \end{cases} $$ where $\alpha$ is a positive integer constant.
As you can see the only difficulty is the indicator $[d \le \alpha h]$. But for me it's a real pain. Please help!