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I am trying to understand a result involving the power of a series that occurs in Gradstein and Ryzhik's Table of Integrals, Series, and Products. Result 0.314 (p.17, 7th ed.) is:

$$\left(\sum_{k=0}^\infty a_k x^k\right)^n=\sum_{k=0}^\infty c_k x^k$$

where $$c_0 = a_0^n, c_m=\frac 1 {ma_0} \sum_{k=1}^m (kn-m+k) a_k c_{m-k}$$ for $m\ge1$ and $n\in\mathbb N$.

What is an appropriate combinatorial interpretation of this result?

One way I am trying to understand it is to see how it arises from the multinomial expansion

$$\left(\sum_{k=0}^\infty b_k \right)^n = \sum_{\kappa\vdash k} \binom k \kappa b^\kappa$$

which has the usual nice combinatorial interpretation of how to put objects in bins. This is suggested in the reference given in Gradstein and Ryzhik, which is an even older book: Smithsonian mathematical formulae and tables of elliptic functions, p.118. However, it feels very much like the the additional structure provided by regrouping powers of $x$ after substituting $b_k = a_k x^k$ has must surely have some significant, nontrivial and well-known combinatorial meaning as well implications that I am simply unaware of. (I hope this is clear; the multiindex notation is new to me and I don't know a nice way to write the result of this last step.)

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Combinatorial interpretation of the power of a series

I am trying to understand a result involving the power of a series that occurs in Gradstein and Ryzhik's Table of Integrals, Series, and Products. Result 0.314 (p.17, 7th ed.) is:

$$\left(\sum_{k=0}^\infty a_k x^k\right)^n=\sum_{k=0}^\infty c_k x^k$$

where $$c_0 = a_0^n, c_m=\frac 1 {ma_0} \sum_{k=1}^m (kn-m+k) a_k c_{m-k}$$ for $m\ge1$ and $n\in\mathbb N$.

What is an appropriate combinatorial interpretation of this result?

One way I am trying to understand it is to see how it arises from the multinomial expansion

$$\left(\sum_{k=0}^\infty b_k \right)^n = \sum_{\kappa\vdash k} \binom k \kappa b^\kappa$$

which has the usual nice combinatorial interpretation of how to put objects in bins. This is suggested in the reference given in Gradstein and Ryzhik, which is an even older book: Smithsonian mathematical formulae and tables of elliptic functions, p.118. However, it feels very much like the the additional structure provided by regrouping powers of $x$ after substituting $b_k = a_k x^k$ has some significant, nontrivial and well-known combinatorial meaning as well that I am simply unaware of. (I hope this is clear; the multiindex notation is new to me and I don't know a nice way to write the result of this last step.)