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How can I reconstruct equation, which solution is given by series $\frac{x^{x-2}}{x!}$? The reverse task is straightforward with Solve function.

It seems pretty close to Lambert W function, but not exactly it.

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    $\begingroup$ Do you mean that you want the generating function of $\frac{k^{k-2}}{k!}$? If so: Exp[-ProductLog[-x]] (1 + ProductLog[-x]/2). $\endgroup$ – J. M. will be back soon Nov 7 '17 at 21:31
  • $\begingroup$ @J.M., thanks, it is. Can you please specify how you received it? I tried GeneratingFunction[n^(n - 2)/n!, n, x], but it failed with "Infinite expression 1/0 encountered". $\endgroup$ – Hasek Nov 8 '17 at 19:43
  • $\begingroup$ I had to derive that result semi-manually since Mathematica does not know about the series expansion for powers of $W(x)/x$. $\endgroup$ – J. M. will be back soon Nov 9 '17 at 1:23

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