Tabulation of
power | as sum | factored |
0 | n | n |
1 | n(n+1)/2 | |
2 | n(n+1)(2n+1)/6 | |
3 | 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:
For odd r>=3:
$latex S_r = S_3*R_{r-2}$ where is a polynomial of degree k.
For even r>=2:
where is a polynomial of degree k.