Series of inverse function $A(s) = \sum_{k>0}a_ks^k$ and $A(s)+A(s)^3=s$.
I want calculate $a_5$. What ways to do it most efficiently?

$A(s) = \sum_{k>0}a_ks^k$ and $A(s)+A(s)^{3} =s$
We know (Cauchy product): $$A(s)^{2} = \sum_{n>0}^{\infty} \left( \sum_{i=0}^{n}a_{i} a_{n-i} \right) s^n$$
And
$$A(s)^{3} = \sum_{n>0}^{\infty} \left( \sum_{j=0}^{n}a_{n-j} \left( \sum_{i=0}^{j}a_{i} a_{j-i} \right) \right) s^n$$
Hence:
$$ \sum_{n>0}^{\infty}a_ns^n+\sum_{n>0}^{\infty} \left( \sum_{j=0}^{n}a_{n-j} \left( \sum_{i=0}^{j}a_{i} a_{j-i} \right) \right) s^n = s$$
For $n=0$:$$a_{0}+{a_0}^{3}=0$$
For n=1:
$$a_{1}+a_{1}.0+0=1$$
By raising the $n$ you get the $a_{n}$
