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After reading this post and the general solution for that case, I wonder if there is a closed form for the general solution for this sum:

$ \sum_{a_1=0}^\infty~\sum_{a_2=0}^\infty~\cdots~\sum_{a_n=0}^\infty \dfrac1{(a_1!+a_2!+\ldots+a_n!)} $

Any idea?

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    $\begingroup$ Did you have any reason to believe that there exists a closed form? This site is not for posting wild conjectures. $\endgroup$
    – jjagmath
    Commented May 14 at 11:29
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    $\begingroup$ If you post this on MathOverflow, they might explain why it's so hard. For now, here's a taste of why. We can equivalently restate this as $\int_0^1\frac{f^n}{x}dx$ with $f=\sum_{i=0}^\infty x^{i!}$. Even numerically, it's hard enough to understand $f$, let alone evaluate such integrals. $\endgroup$
    – J.G.
    Commented May 14 at 11:40
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    $\begingroup$ @jjagmath I Just wondered after reading the original post $\endgroup$
    – user967210
    Commented May 14 at 12:13
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    $\begingroup$ @J.G. thank you for your kind explany, I'll surely do $\endgroup$
    – user967210
    Commented May 14 at 12:14
  • $\begingroup$ Also posted to MathOverflow, mathoverflow.net/questions/471421/… $\endgroup$
    – Gerry Myerson
    Commented May 17 at 2:50

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