# Table of Sums of Powers of the Integers

## Tabulation of $S_r(n) \equiv \sum_{i=1}^n i^r$

 power as sum factored 0 n n 1 $\frac{1}{2}n^2 + \frac{1}{2}n$ n(n+1)/2 2 $\frac{1}{3}n^3 +\frac{1}{2}n^2+ \frac{1}{6}n$ n(n+1)(2n+1)/6 3 $\frac{1}{4}n^4 + \frac{1}{2}n^3+ \frac{1}{4}n^2$ n2(n+1)2/4 4 1/5n5 + 1/2n4 + 1/3n3 − 1/30n n(n+1)(2n+1)(3n2+3n-1)/(6*5) 5 1/6n6 + 1/2n5 + 5/12n4− 1/12 n2 n2(n+1)2 (2n2+2n-1)/(4*3) 6 1/7n7 + 1/2n6 + 1/2n5 − 1/6  n3 + 1/42 n n(n+1)(2n+1)(3n4+6n3-3n+1)/(6*7) 7 1/8n8 + 1/2n7 + 7/12n6− 7/24 n4 + 1/12 n2 n2(n+1)2(3n4+6n3-n2-4n+2)/(4*6) 8 1/9n9 + 1/2n8 + 2/3n7 – 7/15 n5 + 2/9  n3 − 1/30 n n(n+1)(2n+1)(5n6+15n5+5n4-15n3-n2+9n-3)/(6*15) 9 1/10n10+1/2n9 + 3/4n8 – 7/10 n6 + 1/2n4 – 3/20n2 n2(n+1)2(n2+n-1)(2n4+4n3-n2-3n+3)/(4*5) 10 1/11n11+1/2n10+ 5/6n9 –      n7 +  n5 – 1/2n3 + 5/66 n n(n+1)(2n+1)(n2+n-1)(3n6+9n5+2n4-11n3+3n2+10n-5)/(6

Some convenient relationships exist:
$S_3 = S_1^2$
For odd r>=3:
$latex S_r = S_3*R_{r-2}$ where $R_k$ is a polynomial of degree k.
For even r>=2:
$S_r = S_2*T_{r-2}$ where $T_k$ is a polynomial of degree k.