"Counting in their heads" - 1895 oil painting

[Albert Chan]

I was wrong. There seems to be a pattern to sum of powers formula after all ...

\( S_p = n c^p + \large {n^3-n \over 12} \left(
\binom{p}{2} c^{p-2} +
{3n^2-7 \over 20} \binom{p}{4} c^{p-4} +
{3n^4-18n^2+31 \over 112} \binom{p}{6} c^{p-6} +
{5n^6-55n^4+239n^2-381 \over 960} \binom{p}{8} c^{p-8} +
\cdots \right) \)

Example:

50^5 + 51^5 + 52^5 + ... + 150^5              // p=5, c=100, n=101

= 101*100^5 + 102*101*100/12 * (10*100^3 + (3*101^2-7)/20*5*100)
= 1934166665000