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.

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