We would like to identify $s_n$ (non-increasing series) once we know, assuming are all of them convergent:
$$S_k=\sum_{n>=0}{s_n^k}$$
Known for all k values.
As example $\zeta(2k)$ should retrieve $1/{(n+1)^2}$ Another example $\frac{1}{1-a^k}$ should retrieve $a^n$
Consider the tempered distribution $T = \sum_{n \ge 0} \left( \delta_{s_n} - \delta_0 \right)$ where $\delta$ is the Dirac delta. Its Fourier transform is $$ f(t) = \sum_{n \ge 0} \left( \exp(-i t s_n)-1 \right) = \sum_{k=1}^\infty \frac{(-it)^k}{k!} S_k$$ assuming that converges absolutely. In principle you should then be able to recover $T$, and thus the $s_n$, from $f$.