We have $a(0)=a(1)=1$, $$a(n)=(-1)^{n-1}+2\sum\limits_{k=1}^{n-1}\binom{n}{k}(-1)^{n-k-1}a(k)$$ $$1,1,3,13,75,541,4683,\cdots$$ which has nice closed forms, ex. $$a(n)=\sum\limits_{k=0}^{n}k!{n\brace k}$$ Related sequence $b(0)=b(1)=1$, $$b(n)=1+\sum\limits_{k=1}^{n-1}\left[\binom{n}{k}-2^{n-k-1}\right]b(k)$$ $$1,1,2,6,27,166,1282,\cdots$$ unknown for OEIS.
Is there a nice closed form for it?