Sums of integer powers | ||
---|---|---|
Sk(n) = 1k + 2k + 3k + . . . + nk | ||
Exponent (k) | Sum (Sk(n)) | Shortening with N=n(n+1) and M=2n+1 |
1 | n(n+1)/2 | N/2 |
2 | n(n+1)(2n+1)/6 | MN/6 |
3 | n2(n+1)2/4 | N2/4 |
4 | n(n+1)(2n+1)(3n(n+1)-1)/30 | MN(3N-1)/30 |
5 | n2(n+1)2(2n(n+1)-1)/12 | N2(2N-1)/12 |
6 | n(n+1)(2n+1)(3n2(n+1)2-3n(n+1)+1)/42 | MN(3N2-3N+1)/42 |
7 | n2(n+1)2(3n2(n+1)2-4n(n+1)+2)/24 | N2(3N2-4N+2)/24 |
Exponent (k) | Sum (Sk(n)) ..... Shortening with N=n(n+1) and M=2n+1 | |
8 | MN(5N3-10N2+9N-3)/90 | |
9 | N2(2N3-5N2+6N-3)/20 | |
10 | MN(3N4-10N3+17N2-15N+5)/66 | |
11 | N2(2N4-8N3+17N2-20N+10)/24 | |
12 | MN(105N5-525N4+1435N3-2360N2+2073N-691)/2730 | |
13 | N2(30N5-175N4+574N3-1180N2+1382N-691)/420 | |
14 | MN(3N6-21N5+84N4-220N3+359N2-315N+105)/90 | |
15 | N2(3N6-24N5+112N4-352N3+718N2-840N+420)/48 | |
16 | MN(15N7-140N6+770N5-2930N4+7595N3-12370N2+10851N-3617)/510 | |
17 | N2(10N7-105N6+660N5-2930N4+9114N3-18555N2+21702N-10851)/180 | |
18 | MN(105N8-1260N7+9114N6-47418N5+178227N4-460810N3+750167N2-658005N+219335)/3990 | |
19 | N2(42N8-560N7+4557N6-27096N5+118818N4-368648N3+750167N2-877340N+438670)/840 | |
20 | MN(165N9-2475N8+22770N7-155100N6+795795N5-2981895N4+7704835N3-12541460N2+11000493N-3666831)/6930 | |
21 | N2(30N9-495N8+5060N7-38775N6+227370N5-993965N4+3081934N3-6270730N2+7333662N-3666831)/660 | |
22 | MN(15N10-275N9+3135N8-27060N7+181533N6-928151N5+3475154N4-8977920N3+14613279N2-12817695N+4272565)/690 | |
23 | N2(30N10-600N9+7524N8-72160N7+544599N6-3182232N5+13900616N4-43094016N3+87679674N2-102541560N+51270780)/720 | |
24 | MN(273N11-6006N10+83083N9-885885N8+7522515N7-50269856N6+256794972N5-961297596N4+2483374425N3-4042136250N2+3545461365N-1181820455)/13650 | |
25 | N2(42N11-1001N10+15106N9-177177N8+1671670N7-12567464N6+73369992N5-320432532N4+993349770N3-2021068125N2+2363640910N-1181820455)/1092 | |
26 | MN(7N12-182N11+3003N10-38753N9+406120N8-3434184N7+22926780N6-117091548N5+438304419N4-1132285290N3+1842993525N2-1616536467N+538845489)/378 | |
27 | N2(2N12-56N11+1001N10-14092N9+162448N8-1526304N7+11463390N6-66909456N5+292202946N4-905828232N3+1842993525N2-2155381956N+1077690978)/56 | |
28 | MN(15N13-455N12+8827N11-135564N10+1718002N9-17924270N8+151408620N7-1010562744N6+5160854643N5-19318200399N4+49905176965N3-81229418480N2+71248383087N-23749461029)/870 | |
29 | N2(2N13-65N12+1358N11-22594N10+312364N9-3584854N8+33646360N7-252640686N6+1474529898N5-6439400133N4+19962070786N3-40614709240N2+47498922058N-23749461029)/60 | |
30 | MN(231N14-8085N13+182182N12-3283280N11+49483434N10-624177554N9+6504825250N8-54932452344N7+366618940611N6-1872264497565N5+7008271631088N4-18104627279060N3+29468449283827N2-25847523828015N+8615841276005)/14322 | |
31 | N2(231N14-8624N13+208208N12-4040960N11+65977912N10-907894624N9+10407720400N8-97657693056N7+733237881222N6-4279461708720N5+18688724349568N4-57934807292992N3+117873797135308N2-137853460416080N+68926730208040)/7392 | |
32 | MN(1785N15-71400N14+1849260N13-38644060N12+683022158N11-10243967916N10+129062405978N9-1344632829430N8+11354500437885N7-75778715831676N6+386988478042362N5-1448576481258186N4+3742139941405445N3-6090987469844470N2+5342559481563381N-1780853160521127)/117810 | |
33 | N2(210N15-8925N14+246568N13-5520580N12+105080332N11-1707327986N10+23465891996N9-268926565886N8+2523222319530N7-18944678957919N6+110568136583532N5-482858827086062N4+1496855976562178N3-3045493734922235N2+3561706321042254N-1780853160521127)/7140 | |
34 | MN(3N16-136N15+4012N14-96220N13+1970878N12-34658988N11+519169562N10-6538986454N9+68121373265N8-575228568156N7+3838998837626N6-19605069959594N5+73385745287261N4-189579025631030N3+308573030254741N2-270657225128535N+90219075042845)/210 | |
35 | N2(2N16-96N15+3009N14-76976N13+1689324N12-31992912N11+519169562N10-7133439768N9+81745647918N8-766971424208N7+5758498256439N6-33608691359304N5+146771490574522N4-454989661514472N3+925719090764223N2-1082628900514140N+541314450257070)/72 |
Authors: | ||
Gerzson Keri keri@oplab.sztaki.hu (*-15) |
Barbara Keri kerib@ludens.elte.hu (16-17) |
Balazs Molnar mbalazs@mail.scala-rnd.com (18-20) |
Jozsef Ureczky http://www.angelfire.com/id/ureczky/ (21-35) |