Solving a non-linear recurrence relation with binomial coefficient

Question

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.

Answer 1

These are the Bernoulli numbers.

See https://en.wikipedia.org/wiki/Bernoulli_number