First acknowledging the helpful response to the earlier version of this sequence, I have found a complete expression for the sequence in closed form, and would be interested in improvements to the base 2 notation of the sum. Is there a standard form for such notation? Does it depend on the context? The sequence is:
$S_n = \left(\frac{1}{n!}\right) \sum_{i_1i_2...i_{n-1} =\ 00...0}^{11...1} \frac{(n-1)!}{(n-1)^{i_1}(n-2)^{i_2},..., 2^{i_{n-2}}1^{i_{n-1}}}{\big[\ n^{i_1},(n-1)^{i_2},...,2^{i_{n-1}}\big]}$
The $i_n$ to the right of the summation sign are digits of a base-$2$ number, and those same digits $(0,1)$ appear as exponents in the expression. The numeral $1$ in the denominator is an awkward convenience, and I'm sure there's a more elegant way of expressing this.
On the other hand (up to typos) it seems to work well. What I would really like is to put this in recursive form, because the list in brackets, multiplied together term by term, gives (half) the coefficients for $S_{n+1}$. So I think it lends itself to a form like $S_{n+1}= f(S_n)$.
The list in brackets, $\big[n^{1_1},...,\text{ etc.}\big]$, is a somewhat arbitrary property of each term $-$ its salient feature is providing a new coefficient for $S_{n+1}$. In terms of the earlier post, I have excluded from the lists the negations, ($n'$, etc.), which do not help with coefficients for $S_{n+1}$ in any event..
If someone knows the general family to which such a distribution belongs that would interest me. Also, I'd be interested in seeing a good Mathematica routine for this. Thanks.