Power Sums: Large Coefficients

Calculus Level 5

All power sums have a closed polynomial forms for integral powers. For example,

12+22+32++n2=k=1nk2=n33+n22+n61^2+2^2+3^2+\cdots+n^2=\displaystyle \sum_{k=1}^n k^2=\frac{n^3}{3}+\frac{n^2}{2}+\frac{n}{6}

More generally

1m+2m+3m++nm=k=1nkm=i=1m+1aini1^m+2^m+3^m+\cdots+n^m=\displaystyle \sum_{k=1}^n k^m=\displaystyle \sum_{i=1}^{m+1} a_i n^i

In the case of m=2m=2, a1=16a_1=\frac{1}{6}, a2=12a_2=\frac{1}{2}, and a3=13a_3=\frac{1}{3}.

When m=20m=20, if a5a_5 can be written as pq\dfrac{-p}{q} for coprime positive integers p,qp,q, find p+qp+q.

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