One can derive a recurrence for $g_n$ as $g_n = g_{n-1}+g_{n-1}\frac{1}{n!}$, and letting $g_n = \sum_{j=0}^n a_j x^j$ gives a formula for the characteristic polynomial as $n! a_n x^n - \sum_{j=0}^{n-1}a_j x^j=0$. I feel like this is not a bad start but I can't think of what to do next, any ideas?
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3Any reason to expect that a closed form exists? OEIS A238695 lists the infinite product as "conjectured to be irrational, transcendental and normal, none have been shown". – dxiv Jun 03 '21 at 03:24
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@VIVID The question here is about the (finite) partial products, though. – dxiv Jun 03 '21 at 18:37