Ru83n05 19.08.2021 03:37 $R_n=2^{n+1}n!-n!$. For $R_0$ this works, and we can check inductively that $nR_{n-1}+2^nn!=n(2^n(n-1)!-(n-1)!)+2^nn!=n!2^{n+1}-n!$, as desired.