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\pi_4(k)=378k+179k^2+18k^3+k^4  \end{equation}  From the above we can see that \begin{equation}  \pi_m(k) = k^m + \frac{3m(m+1)}{2}k^{m-1} + \frac{m(m+1)(m+2)(125+27m)}{24}k^{m-2} + \frac{m(m+1)(m+2)(m+3)(118+125m+9m^2)}{16}k^{m-3} + \frac{m(m+1)(m+2)(m+3)(m+4)(296662+141845m+33750m^2+1215m^3)}{5760}k^{m-4} + \cdots  \end{equation}  we can then expect a reduction \begin{equation}  \pi_m(k) = k^m + \sum_{i=1}^{m-1}c_i\left[\prod_{j=0}^{i} (m+j)\right]\sigma_i(m)k^{m-i}