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I'm looking for a short and precise proof of the following identity;

$$\left(\sum_{k=0}^\infty \frac{C_k}{k!}x^k\right)^n=\sum_{k=0}^\infty\left[ \sum_{(j_1+...+j_n=k)}\binom{k}{j_1,...,j_n}\frac{C_{j_1}\cdot...\cdot C_{j_n}}{k!}\right]x^k$$

I've got the idea through brute force of smaller convolutions when $n=2,3,4$ and I've used Mathematica to show that the identity holds for the previous cases. But I was hoping for something compact.

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The coefficient of $x^k$ in $$\left(\sum_{k\ge 0}\frac{C_k}{k!}x^k\right)^n$$ is the sum of all possible products of the form

$$\prod_{i=1}^n\frac{C_{j_i}}{j_i!}\;,$$

where the $j_i$ are non-negative integers whose sum is $k$. Thus, it’s

$$\begin{align*} \sum_{j_1+\ldots+j_n=k}\frac{C_{j_1}\cdot\ldots\cdot C_{j_n}}{j_1!\cdot\ldots\cdot j_n!}&=\sum_{j_1+\ldots+j_n=k}\left(\frac{k!}{j_1!\cdot\ldots\cdot j_n!}\cdot\frac{C_{j_1}\cdot\ldots\cdot C_{j_n}}{k!}\right)\\\\ &=\sum_{j_1+\ldots+j_n=k}\binom{k}{j_1,\ldots,j_n}\frac{C_{j_1}\cdot\ldots\cdot C_{j_n}}{k!}\;. \end{align*}$$

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  • $\begingroup$ I appreciate the help here. I just have question. Outside of just expanding and observing, how do you know that the coefficient is the sum of all possible products of the form above? $\endgroup$ Nov 27 '14 at 2:47
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    $\begingroup$ @Eleven-Eleven: I don’t actually have to do the expansion to realize that $\left(\sum_{k\ge 0}\frac{C_k}{k!}x^k\right)^n$ is the sum of all possible terms of the form $\prod_{i=1}^n\left(\frac{C_{j_i}}{j_i!}x^{j_i}\right)$ with $j_i\in\Bbb N$, and from there it’s immediate. $\endgroup$ Nov 27 '14 at 2:57
  • $\begingroup$ @Eleven-Eleven: You’re welcome! $\endgroup$ Nov 27 '14 at 3:13
  • $\begingroup$ I just realized how simple that idea is...I foiled a couple of trinomials in my head, saw the multiplication and addition withing the coefficients. Thanks again. $\endgroup$ Nov 27 '14 at 3:15
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    $\begingroup$ @Eleven-Eleven: Glad to be of help! Getting that kind of insight is even better than just getting the question answered. $\endgroup$ Nov 27 '14 at 3:29

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