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S_4(n)=16n^3-12n^2+5n-2\\  S_5(n)=32n^4-32n^3+18n^2-10n+3\\  \end{equation}  These are then polynomials, such that $S_k(1)=p_k$. We note that the constant terms are A030018, the coefficients of the generating function \frac{1}{P(x)+1}.  For a general polynomial polynomial, there are relationships  we have \begin{equation} S_k(n)=\sum_{i=1}^{k} a_i n^{i-1}  \end{equation}  In general we may note that for $S_k(n)$ \begin{equation}