Power Sums: Large Coefficients

Calculus Level 5

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

$1^2+2^2+3^2+\cdots+n^2=\displaystyle \sum_{k=1}^n k^2=\frac{n^3}{3}+\frac{n^2}{2}+\frac{n}{6}$

More generally

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

In the case of $m=2$, $a_1=\frac{1}{6}$, $a_2=\frac{1}{2}$, and $a_3=\frac{1}{3}$.

When $m=20$, if $a_5$ can be written as $\dfrac{-p}{q}$ for coprime positive integers $p,q$, find $p+q$.

Tip: This might help :)

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