Gregory–Newton summation identity for signed pancake distances

Let $R_k^B(n)$ denote the number of signed permutations of size $n$ whose burnt-pancake distance from the sorted stack is $k$. Summation identity conjecture. If $k,n\geq1$, then

$$R_k^B(n)=\sum_{j=1}^k\left(\sum_{i=0}^{k-j}(-1)^i\binom{i+j-1}{i}\binom{n}{i+j}\right)R_k^B(j).$$

The identity is observed in the source and is motivated by the assertion that the functions $R_k^B(n)$ are integer-valued polynomials together with the Gregory–Newton interpolation formula. No proof or resolution is supplied, so it remains open.