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[tex]s_{n}=\frac{1}{r_{1}-r_{1}^{n}}}\sum_{k=1}^{n-1}s_{k}B_{n,k}\left(

r_{1},r_{2},...,r_{n-k+1}\right)[/tex]

The B_{n,k} are (nonlinear) polynomials in the variables r_{1}, ..., r_{n,k-n+1} that don't involve the s's. (In fact they are Bell polynomials though I'm not sure it is necessary to know B's to get a formula for s_{n}.)

Initial condition: s_{1}=1.

You can see that s_{n} is equal to some function of the previous s_{k} for k running from 1 through n-1.

I want a closed form formula for s_{n} that does not involve any other s_{k}. I've taken it up to 5 to look for a pattern and it seeeeems like there is some function of the B's that's being iterated to get the result of simplifying and substituting previous values of s_{k}.

Actually, I'm not sure this is a difference equation because the sum is not fixed but increases in complexity as n increases.

Any feedback or inquiries as to how this problem arose are quite welcome and appreciated!