Recurrence Derivative

Question

I have a recurrence relation as follows

$ \left\{ \begin{array}{ll} R_0=H & \mbox{if } n = 0 \\ R_1(s)=sR_0 \hspace{.1cm} A & \mbox{if }n=1\\ R_{n+2}(s)=\frac{s}{n+2}\{ R_{n+1}(s)\hspace{.1cm}A+ s\hspace{.1cm} R_{n}(s)\hspace{.1cm}B\} & \mbox{if }n>= 2\\ \end{array} \right. \tag 1$

And we have defined $R(s)=\sum_{n=0}^{\infty} R_n(s) \tag 3$

Specifications

1. H,A,B are constant matrices with dimension $3 \times 3$

2. R has dimension $ 3 \times 3$

3. We define pth derivative as $\frac{\mathrm{d}^p R(s)}{\mathrm{d} s^p}$. It means for example $,R'(s)= \frac{\mathrm{d} R(s)}{\mathrm{d} s },R''(s)=\frac{\mathrm{d}^2 R(s)}{\mathrm{d} s^2}$

Question

Can we find a general finite expression for $\frac{\mathrm{d}^p R(0)}{\mathrm{d} s^p}$ from the given recursion?.

Remember $s=0$ condition cancels lot of terms. More clearly is there any finite expression for pth derivative in-terms of available constants and s?. I am bit confused with way in which we use recursion for it