$D_n:=$ number of partitions of set with $n$ elements with not more than one singleton subset. Express $D_n$ through $F_n$

Question

IMPORTANT: $F_n$ is amount of partitions where in each partition there are no subsets of size less than 2 (no singletons).

I don't know if it can be helpful but I know how to express Bell's number through $F_n$.

$$B_{n} = F_n + F_{n+1}$$

Update: found one of possible expressions $D_n = F_n + n\cdot F_{n-1}$. Are there any others which wouldn’t use F(n-1) but only F(n)?

Answer 1

Well done, that equation that you find says, fix a singleton and then no singletons or no singletons at all. You can use the expression you have in top. Notice then that $B_{n-1}=F_{n-1}+F_n,$ so $F_{n-1}=B_{n-1}-F_n$ and your equations becomes $$D_n=F_n+n(B_{n-1}-F_n)=n\cdot B_{n-1}-(n-1)F_{n}$$