Let $a(n)$ be A004208. Here $$a(n)=n\prod\limits_{j=1}^{n}(2j-1)-\sum\limits_{i=1}^{n-1}a(i)\prod\limits_{j=1}^{n-i}(2j-1)$$ I conjecture that $$a(n)=R(n-1,0)$$ where $$R(n,q)=2(q+2)R(n-1,q+1)+\sum\limits_{j=0}^{q}R(n-1,j), R(0,q)=1$$ Is there a way to prove it?
I would also like to know if this type of recurrence is known in the literature, where instead of $2(q+2)$ we use an arbitrary function $g(q)$, and also modify the second part to $$m\sum\limits_{j=0}^{q}k^{q-j}R(n-1,j)$$ Here $m,k\in\mathbb{R}$.
