50 Identities of Power Summation

(09/26/2009 - update)

For each positive integer n,

• 1 + 2 + 3 + ... + n = (1/2)n² + (1/2)n .

• 1² + 2² + 3² + ... + n² = (1/3)n³ + (1/2)n² + (1/6)n .

• 1³ + 2³ + 3³ + ... + n³ = (1/4)n⁴ + (1/2)n³ + (1/4 )n² . 

• 1⁴ + 2⁴ + 3⁴ + ... + n⁴ = (1/5)n⁵ + (1/2)n⁴ + (1/3)n³ - (1/30)n .

• 1⁵ + 2⁵ + 3⁵ + ... + n⁵ = (1/6)n⁶ + (1/2)n⁵ + (5/12)n⁴ - (1/12)n² .

• 1⁶ + 2⁶ + 3⁶ + ... + n⁶ = (1/7)n⁷ + (1/2)n⁶ + (1/2)n⁵ - (1/6)n³ + (1/42)n .

• 1⁷ + 2⁷ + 3⁷ + ... + n⁷ = (1/8)n⁸ + (1/2)n⁷ + (7/12)n⁶ - (7/24)n⁴ + (1/12)n² .

• 1⁸ + 2⁸ + 3⁸ + ... + n⁸ = (1/9)n⁹ + (1/2)n⁸ + (2/3)n⁷ - (7/15)n⁵ + (2/9)n³ - (1/30)n .

• 1⁹ + 2⁹ + 3⁹ + ... + n⁹ = (1/10)n¹⁰ + (1/2)n⁹ + (3/4)n⁸ - (7/10)n⁶ + (1/2)n⁴ - (3/20)n² .

• 1¹⁰ + 2¹⁰ + 3¹⁰ + ... + n¹⁰ = (1/11)n¹¹ + (1/2)n¹⁰ + (5/6)n⁹ - n⁷ + n⁵ - (1/2)n³ + (5/66)n .

• 1¹¹ + 2¹¹ + 3¹¹ + ... + n¹¹ = (1/12)n¹² + (1/2)n¹¹ + (11/12)n¹⁰ - (11/8)n⁸ + (11/6)n⁶ - (11/8)n⁴ + (5/12)n² .

• 1¹² + 2¹² + 3¹² + ... + n¹² = (1/13)n¹³ + (1/2)n¹² + n¹¹ - (11/6)n⁹ + (22/7)n⁷ - (33/10)n⁵ + (5/3)n³ - (691/2730)n .

• 1¹³ + 2¹³ + 3¹³ + ... + n¹³ = (1/14)n¹⁴ + (1/2)n¹³ + (13/12)n¹² - (143/60)n¹⁰ + (143/28)n⁸ - (143/20)n⁶ + (65/12)n⁴ - (691/420)n² .

• 1¹⁴ + 2¹⁴ + 3¹⁴ + ... +  n¹⁴ = (1/15)n¹⁵ + (1/2)n¹⁴ + (7/6)n¹³ - (91/30)n¹¹ + (143/18)n⁹ - (143/10)n⁷ + (91/6)n⁵ - (691/90)n³ + (7/6)n .

• 1¹⁵ + 2¹⁵ + 3¹⁵ + ... + n¹⁵ = (1/16)n¹⁶ + (1/2)n¹⁵ + (5/4)n¹⁴ - (91/24)n¹² + (143/12)n¹⁰ - (429/16)n⁸ + (455/12)n⁶ - (691/24)n⁴ + (35/4)n² .

• 1¹⁶ + 2¹⁶ + 3¹⁶ + ... + n¹⁶ = (1/17)n¹⁷ + (1/2)n¹⁶ + (4/3)n¹⁵ - (14/3)n¹³ + (52/3)n¹¹ - (143/3)n⁹ + (260/3)n⁷ - (1382/15)n⁵ + (140/3)n³ - (3617/510)n .

• 1¹⁷ + 2¹⁷ + 3¹⁷ + ... + n¹⁷ = (1/18)n¹⁸ + (1/2)n¹⁷ + (17/12)n¹⁶ - (17/3)n¹⁴ + (221/9)n¹² - (2431/30)n¹⁰ + (1105/6)n⁸ - (11747/45)n⁶ + (595/3)n⁴ - (3617/60)n² .

• 1¹⁸ + 2¹⁸ + 3¹⁸ + ... + n¹⁸ = (1/19)n¹⁹ + (1/2)n¹⁸ + (3/2)n¹⁷ - (34/5)n¹⁵ + (34)n¹³ - (663/5)n¹¹ + (1105/3)n⁹ - (23494/35)n⁷ + (714)n⁵ - (3617/10)n³ + (43867/798)n .

• 1¹⁹ + 2¹⁹ + 3¹⁹ + ... + n¹⁹ = (1/20)n²⁰ + (1/2)n¹⁹ + (19/12)n¹⁸ - (323/40)n¹⁶ + (323/7)n¹⁴ - (4199/20)n¹² + (4199/6)n¹⁰ - (223193/140)n⁸ + (2261)n⁶ - (68723/40)n⁴ + (43867/84)n² .

• 1²⁰ + 2²⁰ + 3²⁰ + ... + n²⁰ = (1/21)n²¹ + (1/2)n²⁰ + (5/3)n¹⁹ - (19/2)n¹⁷ + (1292/21)n¹⁵ - (323)n¹³ + (41990/33)n¹¹ - (223193/63)n⁹ + (6460)n⁷ - (68723/10)n⁵ + (219335/63)n³ - (174611/330)n .

• 1²¹ + 2²¹ + 3²¹ + ... + n²¹ = (1/22)n²² + (1/2)n²¹ + (7/4)n²⁰ - (133/12)n¹⁸ + (323/4)n¹⁶ - (969/2)n¹⁴ + (146965/66)n¹² - (223193/30)n¹⁰ + (33915/2)n⁸ - (481061/20)n⁶ + (219335/12)n⁴ - (1222277/220)n² .

• 1²² + 2²² + 3²² + ... + n²² = (1/23)n²³ + (1/2)n²² + (11/6)n²¹ - (77/6)n¹⁹ + (209/2)n¹⁷ - (3553/5)n¹⁵ + (11305/3)n¹³ - (223193/15)n¹¹ + (124355/3)n⁹ - (755953/10)n⁷ + (482537/6)n⁵ - (1222277/30)n³ + (854513/138)n .

• 1²³ + 2²³ + 3²³ + ... + n²³ = (1/24)n²⁴ + (1/2)n²³ + (23/12)n²² - (1771/120)n²⁰ + (4807/36)n¹⁸ - (81719/80)n¹⁶ + (37145/6)n¹⁴ - (5133439/180)n¹² + (572033/6)n¹⁰ - (17386919/80)n⁸ + (11098351/36)n⁶ - (28112371/120)n⁴ + (854513/12)n² .

• 1²⁴ + 2²⁴ + 3²⁴ + ... + n²⁴ = (1/25)n²⁵ + (1/2)n²⁴ + (2)n²³ - (253/15)n²¹ + (506/3)n¹⁹ - (14421/10)n¹⁷ + (29716/3)n¹⁵ - (10266878/195)n¹³ + (208012)n¹¹ - (17386919/30)n⁹ + (22196702/21)n⁷ - (28112371/25)n⁵ + (1709026/3)n³ - (236364091/2730)n .

• 1²⁵ + 2²⁵ + 3²⁵ + ... + n²⁵ = (1/26)n²⁶ + (1/2)n²⁵ + (25/12)n²⁴ - (115/6)n²² + (1265/6)n²⁰ - (24035/12)n¹⁸ + (185725/12)n¹⁶ - (25667195/273)n¹⁴ + (1300075/3)n¹² - (17386919/12)n¹⁰ + (277458775/84)n⁸ - (28112371/6)n⁶ + (21362825/6)n⁴ - (1181820455/1092)n² .

• 1²⁶ + 2²⁶ + 3²⁶ + ... + n²⁶ = (1/27)n²⁷ + (1/2)n²⁶ + (13/6)n²⁵ - (65/3)n²³ + (16445/63)n²¹ - (16445/6)n¹⁹ + (142025/6)n¹⁷ - (10266878/63)n¹⁵ + (2600150/3)n¹³ - (20548177/6)n¹¹ + (3606964075/378)n⁹ - (52208689/3)n⁷ + (55543345/3)n⁵ - (1181820455/126)n³ + (8553103/6)n .

• 1²⁷ + 2²⁷ + 3²⁷ + ... + n²⁷ = (1/28)n²⁸ + (1/2)n²⁷ + (9/4)n²⁶ - (195/8)n²⁴ + (4485/14)n²² - (29601/8)n²⁰ + (142025/4)n¹⁸ - (15400317/56)n¹⁶ + (1671525)n¹⁴ - (61644531/8)n¹² + (721392815/28)n¹⁰ - (469878201/8)n⁸ + (166630035/2)n⁶ - (3545461365/56)n⁴ + (76977927/4)n² .

• 1²⁸ + 2²⁸ + 3²⁸ + ... + n²⁸ = (1/29)n²⁹ + (1/2)n²⁸ + (7/3)n²⁷ - (273/10)n²⁵ + (390)n²³ - (9867/2)n²¹ + (52325)n¹⁹ - (905901/2)n¹⁷ + (3120180)n¹⁵ - (33193209/2)n¹³ + (65581165)n¹¹ - (365460823/2)n⁹ + (333260070)n⁷ - (709092273/2)n⁵ + (179615163)n³ - (23749461029/870)n .

• 1²⁹ + 2²⁹ + 3²⁹ + ... + n²⁹ = (1/30)n³⁰ + (1/2)n²⁹ + (29/12)n²⁸ - (609/20)n²⁶ + (1885/4)n²⁴ - (26013/4)n²² + (303485/4)n²⁰ - (8757043/12)n¹⁸ + (22621305/4)n¹⁶ - (137514723/4)n¹⁴ + (1901853785/12)n¹² - (10598363867/20)n¹⁰ + (4832271015/4)n⁸ - (6854558639/4)n⁶ + (5208839727/4)n⁴ - (23749461029/60)n² .

• 1³⁰ + 2³⁰ + 3³⁰ + ... + n³⁰ = (1/31)n³¹ + (1/2)n³⁰ + (5/2)n²⁹ - (203/6)n²⁷ + (1131/2)n²⁵ - (16965/2)n²³ + (216775/2)n²¹ - (2304485/2)n¹⁹ + (19959975/2)n¹⁷ - (137514723/2)n¹⁵ + (731482225/2)n¹³ - (31795091601/22)n¹¹ + (8053785025/2)n⁹ - (102818379585/14)n⁷ + (15626519181/2)n⁵ - (23749461029/6)n³ + (8615841276005/14322)n .

• 1³¹ + 2³¹ + 3³¹ + ... + n³¹ = (1/32)n³² + (1/2)n³¹ + (31/12)n³⁰ - (899/24)n²⁸ + (2697/4)n²⁶ - (175305/16)n²⁴ + (6720025/44)n²² - (14287807/8)n²⁰ + (68751025/4)n¹⁸ - (4262956413/32)n¹⁶ + (22675948975/28)n¹⁴ - (328549279877/88)n¹² + (49933467155/4)n¹⁰ - (3187369767135/112)n⁸ + (161474031537/4)n⁶ - (736233291899/24)n⁴ + (8615841276005/924)n² .

• 1³² + 2³² + 3³² + ... + n³² = (1/33)n³³ + (1/2)n³² + (8/3)n³¹ - (124/3)n²⁹ + (7192/9)n²⁷ - (70122/5)n²⁵ + (2337400/11)n²³ - (57151228/21)n²¹ + (28947800)n¹⁹ - (4262956413/17)n¹⁷ + (36281518360/21)n¹⁵ - (101092086116/11)n¹³ + (36315248840)n¹¹ - (2124913178090/21)n⁹ + (184541750328)n⁷ - (2944933167596/15)n⁵ + (68926730208040/693)n³ - (7709321041217/510)n .

• 1³³ + 2³³ + 3³³ + ... + n³³ = (1/34)n³⁴ + (1/2)n³³ + (11/4)n³² - (682/15)n³⁰ + (19778/21)n²⁸ - (89001/5)n²⁶ + (292175)n²⁴ - (28575614/7)n²² + (47763870)n²⁰ - (15630840181/34)n¹⁸ + (49887087745/14)n¹⁶ - (21662589882)n¹⁴ + (99866934310)n¹² - (2337404495899/7)n¹⁰ + (761234720103)n⁸ - (16197132421778/15)n⁶ + (17231682552010/21)n⁴ - (84802531453387/340)n² .

• 1³⁴ + 2³⁴ + 3³⁴ + ... + n³⁴ = (1/35)n³⁵ + (1/2)n³⁴ + (17/6)n³³ - (748/15)n³¹ + (23188/21)n²⁹ - (336226/15)n²⁷ + (397358)n²⁵ - (42242212/7)n²³ + (77331980)n²¹ - (822675799)n¹⁹ + (49887087745/7)n¹⁷ - (245509351996/5)n¹⁵ + (261190443580)n¹³ - (7224704805506/7)n¹¹ + (2875775609278)n⁹ - (78671786048636/15)n⁷ + (117175441353668/21)n⁵ - (84802531453387/30)n³ + (2577687858367/6)n .

• 1³⁵ + 2³⁵ + 3³⁵ + ... + n³⁵ = (1/36)n³⁶ + (1/2)n³⁵ + (35/12)n³⁴ - (1309/24)n³² + (11594/9)n³⁰ - (168113/6)n²⁸ + (534905)n²⁶ - (52802765/6)n²⁴ + (123028150)n²² - (5758730593/4)n²⁰ + (249435438725/18)n¹⁸ - (429641365993/4)n¹⁶ + (652976108950)n¹⁴ - (18061762013765/6)n¹² + (10065214632473)n¹⁰ - (137675625585113/6)n⁸ + (292938603384170/9)n⁶ - (593617720173709/24)n⁴ + (90219075042845/12)n² .

• 1³⁶ + 2³⁶ + 3³⁶ + ... + n³⁶ = (1/37)n³⁷ + (1/2)n³⁶ + (3)n³⁵ - (119/2)n³³ + (1496)n³¹ - (34782)n²⁹ + (2139620/3)n²⁷ - (63363318/5)n²⁵ + (192565800)n²³ - (2468027397)n²¹ + (498870877450/19)n¹⁹ - (227457193761)n¹⁷ + (1567142661480)n¹⁵ - (108370572082590/13)n¹³ + (32940702433548)n¹¹ - (275351251170226/3)n⁹ + (1171754413536680/7)n⁷ - (1780853160521127/10)n⁵ + (90219075042845)n³ - (26315271553053477373/1919190)n .

• 1³⁷ + 2³⁷ + 3³⁷ + ... + n³⁷ = (1/38)n³⁸ + (1/2)n³⁷ + (37/12)n³⁶ - (259/4)n³⁴ + (6919/4)n³² - (214489/5)n³⁰ + (2827355/3)n²⁸ - (1172221383/65)n²⁶ + (296872275)n²⁴ - (8301546699/2)n²² + (1845822246565/38)n²⁰ - (935101796573/2)n¹⁸ + (7248034809345/2)n¹⁶ - (2004855583527915/91)n¹⁴ + (101567165836773)n¹² - (5093998146649181/15)n¹⁰ + (5419364162607145/7)n⁸ - (21963855646427233/20)n⁶ + (3338105776585265/4)n⁴ - (26315271553053477373/103740)n² .

• 1³⁸ + 2³⁸ + 3³⁸ + ... + n³⁸ = (1/39)n³⁹ + (1/2)n³⁸ + (19/6)n³⁷ - (703/10)n³⁵ + (11951/6)n³³ - (262922/5)n³¹ + (3704810/3)n²⁹ - (14848137518/585)n²⁷ + (451245858)n²⁵ - (6857799447)n²³ + (1845822246565/21)n²¹ - (935101796573)n¹⁹ + (8100744786915)n¹⁷ - (5078967478270718/91)n¹⁵ + (296888638599798)n¹³ - (17597448142969898/15)n¹¹ + (205935838179071510/63)n⁹ - (59616179611731061/10)n⁷ + (12684801951024007/2)n⁵ - (26315271553053477373/8190)n³ + (2929993913841559/6)n .

• 1³⁹ + 2³⁹ + 3³⁹ + ... + n³⁹ = (1/40)n⁴⁰ + (1/2)n³⁹ + (13/4)n³⁸ - (9139/120)n³⁶ + (9139/4)n³⁴ - (5126979/80)n³² + (4816253/3)n³⁰ - (7424068759/210)n²⁸ + (676868787)n²⁶ - (89151392811/8)n²⁴ + (2181426291395/14)n²² - (36468970066347/20)n²⁰ + (35103227409965/2)n¹⁸ - (7618451217406077/56)n¹⁶ + (827046921813723)n¹⁴ - (114383412929304337/30)n¹² + (267716589632792963/21)n¹⁰ - (2325031004857511379/80)n⁸ + (164902425363312091/4)n⁶ - (26315271553053477373/840)n⁴ + (38089920879940267/4)n² .

• 1⁴⁰ + 2⁴⁰ + 3⁴⁰ + ... + n⁴⁰ = (1/41)n⁴¹ + (1/2)n⁴⁰ + (10/3)n³⁹ - (247/3)n³⁷ + (18278/7)n³⁵ - (155363/2)n³³ + (6214520/3)n³¹ - (1024009484/21)n²⁹ + (3008305720/3)n²⁷ - (89151392811/5)n²⁵ + (1896892427300/7)n²³ - (3473235244414)n²¹ + (36950765694700)n¹⁹ - (2240720946295905/7)n¹⁷ + (2205458458169928)n¹⁵ - (35194896285939796/3)n¹³ + (10708663585311718520/231)n¹¹ - (775010334952503793/6)n⁹ + (235574893376160130)n⁷ - (26315271553053477373/105)n⁵ + (380899208799402670/3)n³ - (261082718496449122051/13530)n .

• 1⁴¹ + 2⁴¹ + 3⁴¹ + ... + n⁴¹ = (1/42)n⁴² + (1/2)n⁴¹ + (41/12)n⁴⁰ - (533/6)n³⁸ + (374699/126)n³⁶ - (374699/4)n³⁴ + (31849415/12)n³² - (20992194422/315)n³⁰ + (4405019090/3)n²⁸ - (281169777327/10)n²⁶ + (19443147379825/42)n²⁴ - (71201322510487/11)n²² + (75749069674135)n²⁰ - (30623186266044035/42)n¹⁸ + (11302974598120881/2)n¹⁶ - (103070767694537974/3)n¹⁴ + (109763801749445114830/693)n¹² - (31775423733052655513/60)n¹⁰ + (4829285314211282665/4)n⁸ - (1078926133675192572293/630)n⁶ + (7808433780387754735/6)n⁴ - (261082718496449122051/660)n² .

• 1⁴² + 2⁴² + 3⁴² + ... + n⁴² = (1/43)n⁴³ + (1/2)n⁴² + (7/2)n⁴¹ - (287/3)n³⁹ + (10127/3)n³⁷ - (1124097/10)n³⁵ + (222945905/66)n³³ - (1354335124/15)n³¹ + (2126560940)n²⁹ - (656062813763/15)n²⁷ + (777725895193)n²⁵ - (130019806323498/11)n²³ + (151498139348270)n²¹ - (1611746645581265)n¹⁹ + (13962498032972853)n¹⁷ - (1442990747723531636/15)n¹⁵ + (16886738730683863820/33)n¹³ - (20220724193760780781/10)n¹¹ + (33804997199478978655/6)n⁹ - (1078926133675192572293/105)n⁷ + (10931807292542856629)n⁵ - (1827579029475143854357/330)n³ + (1520097643918070802691/1806)n .

• 1⁴³ + 2⁴³ + 3⁴³ + ... + n⁴³ = (1/44)n⁴⁴ + (1/2)n⁴³ + (43/12)n⁴² - (12341/120)n⁴⁰ + (22919/6)n³⁸ - (16112057/120)n³⁶ + (563921995/132)n³⁴ - (14559102583/120)n³² + (9144212042/3)n³⁰ - (4030100141687/60)n²⁸ + (2572477961023/2)n²⁶ - (931808611985069/44)n²⁴ + (296109999635255)n²² - (13861021151998879/4)n²⁰ + (66709712824203631/2)n¹⁸ - (15512150538027965087/60)n¹⁶ + (363064882709703072130/231)n¹⁴ - (869491140331713573583/120)n¹² + (290722975915519216433/12)n¹⁰ - (46393823748033280608599/840)n⁸ + (470067713579342835047/6)n⁶ - (78585898267431185737351/1320)n⁴ + (1520097643918070802691/84)n² .

• 1⁴⁴ + 2⁴⁴ + 3⁴⁴ + ... + n⁴⁴ = (1/45)n⁴⁵ + (1/2)n⁴⁴ + (11/3)n⁴³ - (3311/30)n⁴¹ + (38786/9)n³⁹ - (4790071/30)n³⁷ + (16112057/3)n³⁵ - (14559102583/90)n³³ + (12978881608/3)n³¹ - (1528658674433/15)n²⁹ + (56594515142506/27)n²⁷ - (931808611985069/25)n²⁵ + (13028839983951220/23)n²³ - (152471232671987669/21)n²¹ + (77242825375393678)n¹⁹ - (10037273877547506821/15)n¹⁷ + (290451906167762457704/63)n¹⁵ - (735723272588373023801/30)n¹³ + (290722975915519216433/3)n¹¹ - (510332061228366086694589/1890)n⁹ + (1477355671249363195862/3)n⁷ - (78585898267431185737351/150)n⁵ + (16721074083098778829601/63)n³ - (27833269579301024235023/690)n .

• 1⁴⁵ + 2⁴⁵ + 3⁴⁵ + ... + n⁴⁵ = (1/46)n⁴⁶ + (1/2)n⁴⁵ + (15/4)n⁴⁴ - (473/4)n⁴² + (19393/4)n⁴⁰ - (756327/4)n³⁸ + (80560285/12)n³⁶ - (856417799/4)n³⁴ + (24335403015/4)n³² - (1528658674433/10)n³⁰ + (141486287856265/42)n²⁸ - (645098269835817/10)n²⁶ + (48858149939817075/46)n²⁴ - (207915317279983185/14)n²² + (347592714189271551/2)n²⁰ - (10037273877547506821/6)n¹⁸ + (181532441354851536065/14)n¹⁶ - (315309973966445581629/4)n¹⁴ + (1453614879577596082165/4)n¹² - (510332061228366086694589/420)n¹⁰ + (11080167534370223968965/4)n⁸ - (78585898267431185737351/20)n⁶ + (83605370415493894148005/28)n⁴ - (83499808737903072705069/92)n² .

• 1⁴⁶ + 2⁴⁶ + 3⁴⁶ + ... + n⁴⁶ = (1/47)n⁴⁷ + (1/2)n⁴⁶ + (23/6)n⁴⁵ - (253/2)n⁴³ + (10879/2)n⁴¹ - (446039/2)n³⁹ + (50078015/6)n³⁷ - (19697609377/70)n³⁵ + (16961038465/2)n³³ - (1134166113289/5)n³¹ + (112213262782555/21)n²⁹ - (1648584467358199/15)n²⁷ + (1954325997592683)n²⁵ - (207915317279983185/7)n²³ + (380696782207297413)n²¹ - (12150384167557508257/3)n¹⁹ + (245602714774210901735/7)n¹⁷ - (2417376467076082792489/10)n¹⁵ + (2571780171560362299215/2)n¹³ - (1067057946204765453997777/210)n¹¹ + (84947951096838383762065/6)n⁹ - (258210808592988181708439/10)n⁷ + (384584703911271913080823/14)n⁵ - (27833269579301024235023/2)n³ + (596451111593912163277961/282)n .

• 1⁴⁷ + 2⁴⁷ + 3⁴⁷ + ... + n⁴⁷ = (1/48)n⁴⁸ + (1/2)n⁴⁷ + (47/12)n⁴⁶ - (1081/8)n⁴⁴ + (511313/84)n⁴² - (20963833/80)n⁴⁰ + (123877195/12)n³⁸ - (925787640719/2520)n³⁶ + (46892282815/4)n³⁴ - (53305807324583/160)n³² + (1054804670156017/126)n³⁰ - (11069067137976479/60)n²⁸ + (7065640145142777/2)n²⁶ - (3257339970719736565/56)n²⁴ + (1626613523976634401/2)n²² - (571068055875202888079/60)n²⁰ + (11543327594387912381545/126)n¹⁸ - (113616693952575891246983/160)n¹⁶ + (17267666866191004009015/4)n¹⁴ - (50151723471623976337895519/2520)n¹² + (798510740310280807363411/12)n¹⁰ - (12135908003870444540296633/80)n⁸ + (18075481083829779914798681/84)n⁶ - (1308163670227148139046081/8)n⁴ + (596451111593912163277961/12)n² .

• 1⁴⁸ + 2⁴⁸ + 3⁴⁸ + ... + n⁴⁸ = (1/49)n⁴⁹ + (1/2)n⁴⁸ + (4)n⁴⁷ - (2162/15)n⁴⁵ + (47564/7)n⁴³ - (1533939/5)n⁴¹ + (38116060/3)n³⁹ - (50042575174/105)n³⁷ + (16077354108)n³⁵ - (4845982484053/10)n³³ + (272207656814456/21)n³¹ - (1526767881100204/5)n²⁹ + (18841707053714072/3)n²⁷ - (3908807964863683878/35)n²⁵ + (1697335851106053288)n²³ - (326324603357258793188/15)n²¹ + (4860348460794910476440/21)n¹⁹ - (340850081857727673740949/170)n¹⁷ + (13814133492952803207212)n¹⁵ - (100303446943247952675791038/1365)n¹³ + (290367541931011202677604)n¹¹ - (12135908003870444540296633/15)n⁹ + (72301924335319119659194724/49)n⁷ - (7848982021362888834276486/5)n⁵ + (2385804446375648653111844/3)n³ - (5609403368997817686249127547/46410)n .

• 1⁴⁹ + 2⁴⁹ + 3⁴⁹ + ... + n⁴⁹ = (1/50)n⁵⁰ + (1/2)n⁴⁹ + (49/12)n⁴⁸ - (2303/15)n⁴⁶ + (7567)n⁴⁴ - (3579191/10)n⁴² + (93384347/6)n⁴⁰ - (9218369111/15)n³⁸ + (65649195941/3)n³⁶ - (237453141718597/340)n³⁴ + (238181699712649/12)n³² - (37405813086954998/75)n³⁰ + (32972987343999626/3)n²⁸ - (13680827877022893573/65)n²⁶ + (3465394029341525463)n²⁴ - (726813889295712766646/15)n²² + (1701121961278218666754/3)n²⁰ - (5567218003676218671102167/1020)n¹⁸ + (169223135288671839288347/4)n¹⁶ - (50151723471623976337895519/195)n¹⁴ + (3557002388654887232800649/3)n¹² - (594659492189651782474535017/150)n¹⁰ + (18075481083829779914798681/2)n⁸ - (64100019841130258813257969/5)n⁶ + (29226104468101696000620089/3)n⁴ - (39265823582984723803743892829/13260)n² .

• 1⁵⁰ + 2⁵⁰ + 3⁵⁰ + ... + n⁵⁰ = (1/51)n⁵¹ + (1/2)n⁵⁰ + (25/6)n⁴⁹ - (490/3)n⁴⁷ + (75670/9)n⁴⁵ - (416185)n⁴³ + (56941675/3)n⁴¹ - (92183691110/117)n³⁹ + (88715129650/3)n³⁷ - (33921877388371/34)n³⁵ + (541322044801475/18)n³³ - (2413278263674516/3)n³¹ + (56849978179309700/3)n²⁹ - (45602759590076311910/117)n²⁷ + (6930788058683050926)n²⁵ - (316006038824222942020/3)n²³ + (12150871151987276191100/9)n²¹ - (1465057369388478597658465/102)n¹⁹ + (248857551895105646012275/2)n¹⁷ - (100303446943247952675791038/117)n¹⁵ + (13680778417903412433848650/3)n¹³ - (594659492189651782474535017/33)n¹¹ + (451887027095744497869967025/9)n⁹ - (91571456915900369733225670)n⁷ + (292261044681016960006200890/3)n⁵ - (196329117914923619018719464145/3978)n³ + (495057205241079648212477525/66)n.

•  Download (powersum.txt)

• View Series in Summation Notation (17 identities)

(09/26/2009 - Update)

• Finite Odd Power Sum
• Finite Alternative Odd Power Sum
• Some Finite Series Found in Closed-Form
• Repeated Power Sum
• Sum Of Partial Factorials

We all agree that your theory is crazy, but is it crazy enough?
                                                                        Niels Bohr (1885-1962)