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On Wikipedia they claim that $$ \left(\sum_{k=0}^\infty a_k x^k\right)^N $$ is another power series with $c_0 = a_0^n$ and $$c_m = \frac{1}{ma_0} \sum_{k=1}^m (kN-m+k) a_k c_{m-k}.$$ I tried proving this by induction but it's an absolute nightmare. Can someone provide some kind of motivation for where this power series expansion comes from?

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That formula follows from the form of the derivative: $$ c'(x)=N·a(x)^{N-1}·a'(x)\implies a(x)·c'(x)=N·c(x)·a'(x) $$ and then looking at the coefficients of $x^{m-1}$ $$ \sum_{j=1}^m a_{m-j}·(j)c_{j}=N·\sum_{j=0}^{m-1}c_j(m-j)a_{m-j} \implies ma_0·c_m=\sum_{j=0}^{m-1} (Nm-Nj-j)a_{m-j}c_j $$ or set $k=m-j$ to obtain the given formula.

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