I'm looking for an elementary combinatorial/generating function/etc proof of the following result:

For nonnegative integers $r$,

$$\frac{1}{r!} = \sum_{p_0+p_1+\cdots = r} \frac{1}{(p_0!)^2(p_1!)^2\cdots{p_0+p_1+1\choose 1}{p_1+p_2+2\choose 2}{p_2+p_3+3\choose 3}\cdots}.$$

Here the sum is over all sequences of nonnegative integers $(p_0,p_1,...)$ that sum to $r$. (Only finitely many terms in each such sequence will be nonzero.)

It is related to a result of Diaconis and Shahshahani that the trace of a random unitary matrix (with probability measure being given by the Haar measure) is distributed like a Gaussian variable, and indeed can be proven using this result, but I had initially hoped to proceed in the other direction. The above sum, after all, can be evaluated for specific $r$ by inspection (although this rapdily becomes a bit tedious for $r > 2$), and it ought to be possible to somehow summarize this information in a general.

Edit:

Alternatively phrased, we want

$$e^x = \sum_{p_0,p_1,.. = 0}^\infty \frac{x^{p_0+p_1+\cdots}}{\left(\prod_{j \geq 0}(p_j!)^2\right)\cdot\left(\prod_{k\geq 1}{p_{k-1}+p_k+k\choose k}\right)} = \lim_{\lambda \rightarrow \infty} \sum_{p_0,p_1,.. p_\lambda = 0}^\infty \frac{x^{p_0+p_1+\cdots + p_\lambda}}{\left(\prod_{j=0}^\lambda(p_j!)^2\right)\cdot\left(\prod_{k=1}^\lambda{p_{k-1}+p_k+k\choose k}\right){p_\lambda+\lambda+1\choose \lambda+1}}$$