# Solving a non-linear recurrence relation with binomial coefficient

I'm trying to solve a recurrence relation

$$\displaystyle \sum_{i=0}^{n}\binom{n}{i}\frac{A_i}{(n-i+1)}=0$$, where $$A_0=1$$

first few terms are $$A_1=-\frac{1}{2}$$, $$A_2=\frac{1}{6}$$, $$A_3=0$$, $$A_4=-\frac{1}{30}$$, $$A_5=0$$, $$A_6=\frac{1}{42}$$, $$A_7=0$$, $$A_8=-\frac{1}{30}$$ .

It seems like $$A_n=0$$ for odd number $$n$$

but I can't find pattern for even term.

Note that the relation is similar to Catalan number's recursive formula
$$\displaystyle C_n =\sum_{i=0}^{n-1}\binom{2n-2k-1}{n-k-1}\frac{C_i}{n-1}$$ , $$C_0=1$$ whose solution is $$C_n = \frac{1}{n+1}\binom{2n}{n}$$,

Thanks for helping.