| Sums of integer powers with alternating sign | ||
|---|---|---|
| Vk(n) = 1k - 2k + 3k + . . . +(-1)n+1 nk | ||
| Exponent (k) | Sum (Vk(n)) ..... Shortening with N=n(n+1), M=2n+1 and T=[(n+1)/2] | |
| 1 | (-1)n+1T | |
| 2 | (-1)n+1N/2 | |
| 3 | (-1)n+1(MN/4-T/2) | |
| 4 | (-1)n+1N(N-1)/2 | |
| 5 | (-1)n+1(MN(N-2)/4+T) | |
| 6 | (-1)n+1N(N2-3N+3)/2 | |
| 7 | (-1)n+1(MN(2N2-9N+17)/8-17T/4) | |
| 8 | (-1)n+1N(N3-6N2+17N-17)/2 | |
| 9 | (-1)n+1(MN(N3-8N2+33N-62)/4+31T) | |
| 10 | (-1)n+1N(N4-10N3+55N2-155N+155)/2 | |
| 11 | (-1)n+1(MN(2N4-25N3+180N2-736N+1382)/8-691T/2) | |
| 12 | (-1)n+1N(N5-15N4+135N3-736N2+2073N-2073)/2 | |
| 13 | (-1)n+1(MN(N5-18N4+200N3-1424N2+5817N-10922)/4+5461T) | |
| 14 | (-1)n+1N(N6-21N5+280N4-2492N3+13573N2-38227N+38227)/2 | |
| 15 | (-1)n+1(MN(4N6-98N5+1554N4-17045N3+121210N2-495087N+1382)/8-929569T/8) | |
| 16 | (-1)n+1N(N7-28N6+518N5-6818N4+60605N3-330058N2+929569N-929569)/2 | |
| 17 | (-1)n+1(MN(N7-32N6+686N5-10724N4+117455N3-835128N2+3411073N-6404582)/4+3202291T) | |
| 18 | (-1)n+1N(N8-36N7+882N6-16086N5+211419N4-1879038N3+10233219N2-28820619N+28820619)/2 | |
| 19 | (-1)n+1(MN(2N8-81N7+2256N6-47628N5+743346N4-8140201N3+57877532N2-236399964N+443861162)/8-221930581T/2) | |
| 20 | (-1)n+1N(N9-45N8+1410N7-34020N6+619455N5-8140201N4+72346915N3-393999940N2+1109652905N-1109652905)/2 | |
| 21 | (-1)n+1(MN(N9-50N8+1755N7-48096N6+1013607N5-15816822N4+173203030N3-1231486400N2+5029988121N-9444233042)/4+4722116521T) | |
| 22 | (-1)n+1N(N10-55N9+2145N8-66132N7+1592811N6-28997507N5+381046666N4-3386587600N3+18443289777N2-51943281731N+51943281731)/2 | |
| 23-30 | (In the author's form and without supervision) CLICK HERE | |
| Authors: | ||
| Gerzson Keri keri@oplab.sztaki.hu (*-12) |
Jozsef Ureczky http://www.angelfire.com/id/ureczky/ (13-22) |
Gabor Zavarko zavarko@matavnet.hu (23-30) |