自然数幂和公式(1-267)

S(n,1)=(1/2)n^2+(1/2)n^1

S(n,2)=(1/3)n^3+(1/2)n^2+(1/6)n^1

S(n,3)=(1/4)n^4+(1/2)n^3+(1/4)n^2

S(n,4)=(1/5)n^5+(1/2)n^4+(1/3)n^3-(1/30)n^1

S(n,5)=(1/6)n^6+(1/2)n^5+(5/12)n^4-(1/12)n^2

S(n,6)=(1/7)n^7+(1/2)n^6+(1/2)n^5-(1/6)n^3+(1/42)n^1

S(n,7)=(1/8)n^8+(1/2)n^7+(7/12)n^6-(7/24)n^4+(1/12)n^2

S(n,8)=(1/9)n^9+(1/2)n^8+(2/3)n^7-(7/15)n^5+(2/9)n^3-(1/30)n^1

S(n,9)=(1/10)n^10+(1/2)n^9+(3/4)n^8-(7/10)n^6+(1/2)n^4-(3/20)n^2

S(n,10)=(1/11)n^11+(1/2)n^10+(5/6)n^9-n^7+n^5-(1/2)n^3+(5/66)n^1

S(n,11)=(1/12)n^12+(1/2)n^11+(11/12)n^10-(11/8)n^8+(11/6)n^6-(11/8)n^4+(5/12)n^2

S(n,12)=(1/13)n^13+(1/2)n^12+n^11-(11/6)n^9+(22/7)n^7-(33/10)n^5+(5/3)n^3-(691/2730)n^1

S(n,13)=(1/14)n^14+(1/2)n^13+(13/12)n^12-(143/60)n^10+(143/28)n^8-(143/20)n^6+(65/12)n^4-(691/420)n^2

S(n,14)=(1/15)n^15+(1/2)n^14+(7/6)n^13-(91/30)n^11+(143/18)n^9-(143/10)n^7+(91/6)n^5-(691/90)n^3+(7/6)n^1

S(n,15)=(1/16)n^16+(1/2)n^15+(5/4)n^14-(91/24)n^12+(143/12)n^10-(429/16)n^8+(455/12)n^6-(691/24)n^4+(35/4)n^2

S(n,16)=(1/17)n^17+(1/2)n^16+(4/3)n^15-(14/3)n^13+(52/3)n^11-(143/3)n^9+(260/3)n^7-(1382/15)n^5+(140/3)n^3-(3617/510)n^1

S(n,17)=(1/18)n^18+(1/2)n^17+(17/12)n^16-(17/3)n^14+(221/9)n^12-(2431/30)n^10+(1105/6)n^8-(11747/45)n^6+(595/3)n^4-(3617/60)n^2

S(n,18)=(1/19)n^19+(1/2)n^18+(3/2)n^17-(34/5)n^15+34n^13-(663/5)n^11+(1105/3)n^9-(23494/35)n^7+714n^5-(3617/10)n^3+(43867/798)n^1

S(n,19)=(1/20)n^20+(1/2)n^19+(19/12)n^18-(323/40)n^16+(323/7)n^14-(4199/20)n^12+(4199/6)n^10-(223193/140)n^8+2261n^6-(68723/40)n^4+(43867/84)n^2

S(n,20)=(1/21)n^21+(1/2)n^20+(5/3)n^19-(19/2)n^17+(1292/21)n^15-323n^13+(41990/33)n^11-(223193/63)n^9+6460n^7-(68723/10)n^5+(219335/63)n^3-(174611/330)n^1

S(n,21)=(1/22)n^22+(1/2)n^21+(7/4)n^20-(133/12)n^18+(323/4)n^16-(969/2)n^14+(146965/66)n^12-(223193/30)n^10+(33915/2)n^8-(481061/20)n^6+(219335/12)n^4-(1222277/220)n^2

S(n,22)=(1/23)n^23+(1/2)n^22+(11/6)n^21-(77/6)n^19+(209/2)n^17-(3553/5)n^15+(11305/3)n^13-(223193/15)n^11+(124355/3)n^9-(755953/10)n^7+(482537/6)n^5-(1222277/30)n^3+(854513/138)n^1

S(n,23)=(1/24)n^24+(1/2)n^23+(23/12)n^22-(1771/120)n^20+(4807/36)n^18-(81719/80)n^16+(37145/6)n^14-(5133439/180)n^12+(572033/6)n^10-(17386919/80)n^8+(11098351/36)n^6-(28112371/120)n^4+(854513/12)n^2

S(n,24)=(1/25)n^25+(1/2)n^24+2n^23-(253/15)n^21+(506/3)n^19-(14421/10)n^17+(29716/3)n^15-(10266878/195)n^13+208012n^11-(17386919/30)n^9+(22196702/21)n^7-(28112371/25)n^5+(1709026/3)n^3-(236364091/2730)n^1

S(n,25)=(1/26)n^26+(1/2)n^25+(25/12)n^24-(115/6)n^22+(1265/6)n^20-(24035/12)n^18+(185725/12)n^16-(25667195/273)n^14+(1300075/3)n^12-(17386919/12)n^10+(277458775/84)n^8-(28112371/6)n^6+(21362825/6)n^4-(1181820455/1092)n^2

S(n,26)=(1/27)n^27+(1/2)n^26+(13/6)n^25-(65/3)n^23+(16445/63)n^21-(16445/6)n^19+(142025/6)n^17-(10266878/63)n^15+(2600150/3)n^13-(20548177/6)n^11+(3606964075/378)n^9-(52208689/3)n^7+(55543345/3)n^5-(1181820455/126)n^3+(8553103/6)n^1

S(n,27)=(1/28)n^28+(1/2)n^27+(9/4)n^26-(195/8)n^24+(4485/14)n^22-(

29601/8)n^20+(142025/4)n^18-(15400317/56)n^16+1671525n^14-(61644531/8)n^12+(721392815/28)n^10-(469878201/8)n^8+(166630035/2)n^6-(3545461365/56)n^4+(76977927/4)n^2

S(n,28)=(1/29)n^29+(1/2)n^28+(7/3)n^27-(273/10)n^25+390n^23-(9867/2)n^21+52325n^19-(905901/2)n^17+3120180n^15-(33193209/2)n^13+65581165n^11-(365460823/2)n^9+333260070n^7-(709092273/2)n^5+179615163n^3-(23749461029/870)n^1

S(n,29)=(1/30)n^30+(1/2)n^29+(29/12)n^28-(609/20)n^26+(1885/4)n^24-(26013/4)n^22+(303485/4)n^20-(8757043/12)n^18+(22621305/4)n^16-(137514723/4)n^14+(1901853785/12)n^12-(10

598363867/20)n^10+(4832271015/4)n^8-(6854558639/4)n^6+(5208839727/4)n^4-(23749461029/60)n^2

S(n,30)=(1/31)n^31+(1/2)n^30+(5/2)n^29-(203/6)n^27+(1131/2)n^25-(16965/2)n^23+(216775/2)n^21-(2304485/2)n^19+(19959975/2)n^17-(137514723/2)n^15+(731482225/2)n^13-(31795091601/22)n^11+(8053785025/2)n^9-(102818379585/14)n^7+(156********/2)n^5-(23749461029/6)n^3+(8615841276005/14322)n^1

S(n,31)=(1/32)n^32+(1/2)n^31+(31/12)n^30-(899/24)n^28+(2697/4)n^26-(175305/16)n^24+(6720025/44)n^22-(14287807/8)n^20+(68751025/4)n^18-(4262956413/32)n^16+(22675948975/28)n^14-(328549279877/88)n^12+(49933467155/4)n^10-(3187369767135/112)n^8+(161474031537/4)n^6-(736233291899/24)n^4+(8615841276005/924)n^2

S(n,32)=(1/33)n^33+(1/2)n^32+(8/3)n^31-(124/3)n^29+(7192/9)n^27-(70122/5)n^25+(2337400/11)n^23-(57151228/21)n^21+28947800n^19-(4262956413/17)n^17+(36281518360/21)n^15-(101092086116/11)n^13+36315248840n^11-(2124913178090/21)n^9+184541750328n^7-(2944933167596/15)n^5+(68926730208040/693)n^3-(7709321041217/510)n^1

S(n,33)=(1/34)n^34+(1/2)n^33+(11/4)n^32-(682/15)n^30+(19778/21)n^28-(89001/5)n^26+292175n^24-(28575614/7)n^22+47763870n^20-(156********/34)n^18+(49887087745/14)n^16-21662589882n^14+99866934310n^12-(2337404495899/7)n^10+761234720103n^8-(16197132421778/15)n^6+(17231682552010/21)n^4-(84802531453387/340)n^2

S(n,34)=(1/35)n^35+(1/2)n^34+(17/6)n^33-(748/15)n^31+(23188/21)n^29-(336226/15)n^27+397358n^25-(42242212/7)n^23+77331980n^21-822675799n^19+(49887087745/7)n^17-(245509351996/5)n^15+261190443580n^13-(7224704805506/7)n^11+2875775609278n^9-(78671786048636/15)n^7+(117175441353668/21)n^5-(84802531453387/30)n^3+(2577687858367/6)n^1

S(n,35)=(1/36)n^36+(1/2)n^35+(35/12)n^34-(1309/24)n^32+(11594/9)n^30-(168113/6)n^28+534905n^26-(52802765/6)n^24+123028150n^22-(5758730593/4)n^20+(249435438725/18)n^18-(429641365993/4)n^16+652976108950n^14-(18061762013765/6)n^12+10065214632473n^10-(137675625585113/6)n^8+(292938603384170/9)n^6-(593617720173709/24)n^4+(90219075042845/12)n^2

S(n,36)=(1/37)n^37+(1/2)n^36+3n^35-(119/2)n^33+1496n^31-34782n^29+(2139620/3)n^27-(63363318/5)n^25+192565800n^23-2468027397n^21+(498870877450/19)n^19-227457193761n^17+1567142661480n^15-(108370572082590/13)n^13+32940702433548n^11-(275351251170226/3)n^9+(1171754413536680/7)n^7-(1780853160521127/10)n^5+90219075042845n^3-(26315271553053477373/1919190)n^1

S(n,37)=(1/38)n^38+(1/2)n^37+(37/12)n^36-(259/4)n^34+(6919/4)n^32-(214489/5)n^30+(2827355/3)n^28-

(1172221383/65)n^26+296872275n^24-(8301546699/2)n^22+(1845822246565/38)n^20-(935101796573/2)n^18+(7248034809345/2)n^16-(2004855583527915/91)n^14+101567165836773n^12-(5093998146649181/15)n^10+(5419364162607145/7)n^8-(21963855646427233/20)n^6+(3338105776585265/4)n^4-(26315271553053477373/103740)n^2

S(n,38)=(1/39)n^39+(1/2)n^38+(19/6)n^37-(703/10)n^35+(11951/6)n^33-(262922/5)n^31+(3704810/3)n^29-(14848137518/585)n^27+451245858n^25-6857799447n^23+(1845822246565/21)n^21-935101796573n^19+8100744786915n^17-(5078967478270718/91)n^15+2

96888638599798n^13-(17597448142969898/15)n^11+(205935838179071510/63)n^9-(59616179611731061/10)n^7+(12684801951024007/2)n^5-(26315271553053477373/8190)n^3+(2929993913841559/6)n^1

S(n,39)=(1/40)n^40+(1/2)n^39+(13/4)n^38-(9139/120)n^36+(9139/4)n^34-(5126979/80)n^32+(4816253/3)n^30-(7424068759/210)n^28+676868787n^26-(89151392811/8)n^24+(2181426291395/14)n^22-(36468970066347/20)n^20+(35103227409965/2)n^18-(7618451217406077/56)n^16+827046921813723n^14-(114383412929304337/30)n^12+(267716589632792963/21)n^10-(2325031004857511379/80)n^8+(164902425363312091/4)n^6-(26315271553053477373/840)n^4+(38089920879940267/4)n^2

S(n,40)=(1/41)n^41+(1/2)n^40+(10/3)n^39-(247/3)n^37+(18278/7)n^35-(155363/2)n^33+(6214520/3)n^31-(1024009484/21)n^29+(3008305720/3)n^27-(89151392811/5)n^25+(1896892427300/7)n^23-3473235244414n^21+36950765694700n^19-(2240720946295905/7)n^17+2205458458169928n^15-(35194896285939796/3)n^13+(10708663585311718520/231)n^11-(775010334952503793/6)n^9+235574893376160130n^7-(26315271553053477373/105)n^5+(380899208799402670/3)n^3-(261082718496449122051/13530)n^1

S(n,41)=(1/42)n^42+(1/2)n^41+(41/12)n^40-(533/6)n^38+(374699/126)n^36-(374699/4)n^34+(31849415/12)n^32-(20992194422/315)n^30+(4405019090/3)n^28-(281169777327/10)n^26+(19443147379825/42)n^24-(71201322510487/11)n^22+75749069674135n^20-(30623186266044035/42)n^18+(11302974598120881/2)n^16-(103070767694537974/3)n^14+(109763801749445114830/693)n^12-(31775423733052655513/60)n^10+(4829285314211282665/4)n^8-(1078926133675192572293/630)n^6+(7808433780387754735/6)n^4-(261082718496449122051/660)n^2自然数

S(n,42)=(1/43)n^43+(1/2)n^42+(7/2)n^41-(287/3)n^39+(10127/3)n^37-(1124097/10)n^35+(222945905/66)n^33-(1354335124/15)n^31+2126560940n^29-(656062813763/15)n^27+777725895193n^25-(130019806323498/11)n^23+151498139348270n^21-1611746645581265n^19+13962498032972853n^17-(1442990747723531636/15)n^15+(16886738730683863820/33)n^13-(20220724193760780781/10)n^11+(33804997199478978655/6)n^9-(1078926133675192572293/105)n^7+10931807292542856629n^5-(1827579029475143854357/330)n^3+(1520097643918070802691/1806)n^1

S(n,43)=(1/44)n^44+(1/2)n^43+(43/12)n^42-(12341/120)n^40+(22919/6)n^38-(16112057/120)n^36+(563921995/132)n^34-(14559102583/120)n^32+(9144212042/3)n^30-(4030100141687/60)n^28+(257

2477961023/2)n^26-(931808611985069/44)n^24+296109999635255n^22-(13861021151998879/4)n^20+(66709712824203631/2)n^18-(15512150538027965087/60)n^16+(363064882709703072130/231)n^14-(869491140331713573583/120)n^12

+(290722975915519216433/12)n^10-(46393823748033280608599/840)n^8+(470067713579342835047/6)n^6-(78585898267431185737351/1320)n^4+(1520097643918070802691/84)n^2

S(n,44)=(1/45)n^45+(1/2)n^44+(11/3)n^43-(3311/30)n^41+(38786/9)n^39-(4790071/30)n^37+(16112057/3)n^35-(14559102583/90)n^33+(12978881608/3)n^31-(1528658674433/15)n^29+(56594515142506/27)n^27-(931808611985069/25)n^25+(13028839983951220/23)n^23-(152471232671987669/21)n^21+77242825375393678n^19-(10037273877547506821/15)n^17+(290451906167762457704/63)n^15-(735723272588373023801/30)n^13+(290722975915519216433/3)n^11-(510332061228366086694589/1890)n^9+(1477355671249363195862/3)n^7-(78585898267431185737351/150)n^5+(16721074083098778829601/63)n^3-(27833269579301024235023/690)n^1

S(n,45)=(1/46)n^46+(1/2)n^45+(15/4)n^44-(473/4)n^42+(19393/4)n^40-(756327/4)n^38+(80560285/12)

n^36-(856417799/4)n^34+(24335403015/4)n^32-(1528658674433/10)n^30+(141486287856265/42)n^28-(645098269835817/10)n^26+(48858149939817075/46)n^24-(207915317279983185/14)n^22+(347592714189271551/2)n^20-(10037273877547506821/6)n^18+(181532441354851536065/14)n^16-(315309973966445581629/4)n^14+(1453614879577596082165/4)n^12-(510332061228366086694589/420)n^10+(11080167534370223968965/4)n^8-(78585898267431185737351/20)n^6+(83605370415493894148005/28)n^4-(83499808737903072705069/92)n^2

S(n,46)=(1/47)n^47+(1/2)n^46+(23/6)n^45-(253/2)n^43+(10879/2)n^41-(446039/2)n^39+(50078015/6)n^37-(19697609377/70)n^35+(16961038465/2)n^33-(1134166113289/5)n^31+(112213262782555/21)n^29-(1648584467358199/15)n^27+1954325997592683n^25-(207915317279983185/7)n^23+380696782207297413n^21-(12150384167557508257/3)n^19+(245602714774210901735/7)n^17-(2417376467076082792489/10)n^15+(2571780171560362299215/2)n^13-(1067057946204765453997777/210)n^11+(84947951096838383762065/6)n^9-(258210808592988181708439/10)n^7+(384584703911271913080823/14)n^5-(27833269579301024235023/2)n^3+(596451111593912163277961/282)n^1

S(n,47)=(1/48)n^48+(1/2)n^47+(47/12)n^46-(1081/8)n^44+(511313/84)n^42-(20963833/80)n^40+(123877195/12)n^38-(925787640719/2520)n^36+(46892282815/4)n^34-(53305807324583/160)n^32+(1054804670156017/126)n^30-(11069067137976479/60)n^28+(7065640145142777/2)n^26-(3257339970719736565/56)n^24+(1626613523976634401/2)n^22-(571068055875202888079/60)n^20+(11543327594387912381545/126)n^18-(113616693952575891246983/160)n^16+(17267666866191004009015/4)n^14-(50151723471623976337895519/2520)n^12+(798510740310280807363411/12)n^10-(12135908003870444540296633/80)n^8+(18075481083829779914798681/84)n^6-(1308163670227148139046081/8)n^4+(596451111593912163277961/12)n^2

S(n,48)=(1/49)n^49+(1/2)n^48+4n^47-(2162/15)n^45+(47564/7)n^43-(1533939/5)n^41+(38116060/3)n^39-(50042575174/105)n^37+16077354108n^35-(4845982484053/10)n^33+(272207656814456/21)n^31-(1526767881100204/5)n^29+(18841707053714072/3)n^27-(3908807964863683878/35)n^25+1697335851106053288n^23-(326324603357258793188/15)n^21+(4860348460794910476440/21)n^19-(340850081857727673740949/170)n^17+13814133492952803207212n^15-(100303

446943247952675791038/1365)n^13+290367541931011202677604n^11-(121359080038704445402

96633/15)n^9+(72301924335319119659194724/49)n^7-(7848982021362888834276486/5)n^5+(2385804446375648653111844/3)n^3-(5609403368997817686249127547/46410)n^1

S(n,49)=(1/50)n^50+(1/2)n^49+(49/12)n^48-(2303/15)n^46+7567n^44-(3579191/10)n^42+(93384347/6)n^40-(9218369111/15)n^38+(65649195941/3)n^36-(237453141718597/340)n^34+(238181699712649/12)n^32-(37405813086954998/75)n^30+(32972987343999626/3)n^28-(13680827877022893573/65)n^26+3465394029341525463n^24-(726813889295712766646/15)n^22+(1701121961278218666754/3)n^20-(5567218003676218671102167/1020)n^18+(169223135288671839288347/4)n^16-(50151723471623976337895519/195)n^14+(3557002388654887232800649/3)n^12-(594659492189651782474535017/150)n^10+(18075481083829779914798681/2)n^8-(64100019841130258813257969/5)n^6+(29226104468101696000620089/3)n^4-(39265823582984723803743892829/13260)n^2

S(n,50)=(1/51)n^51+(1/2)n^50+(25/6)n^49-(490/3)n^47+(75670/9)n^45-416185n^43+(56941675/3)n^41-(92183691110/117)n^39+(88715129650/3)n^37-(33921877388371/34)n^35+(541322044801475/18)n^33-(2413278263674516/3)n^31+(56849978179309700/3)n^29-(45602759590076311910/117)n^27+6930788058683050926n^25-(316006038824222942020/3)n^23+(12150871151987276191100/9)n^21-(1465057369388478597658465/102)n^19+(2488575518951056

46012275/2)n^17-(100303446943247952675791038/117)n^15+(13680778417903412433848650/3)n^13-(594659492189651782474535017/33)n^11+(451887027095744497869967025/9)n^9-91571456915900369733225670n^7+(292261044681016960006200890/3)n^5-(196329117914923619018719464145/3978)n^3+(495057205241079648212477525/66)n^1

S(n,51)=(1/52)n^52+(1/2)n^51+(17/4)n^50-(4165/24)n^48+(27965/3)n^46-(1929585/4)n^44+(138286925/6)n^42-(156712274887/156)n^40+39688347475n^38-(33921877388371/24)n^36+(541322044801475/12)n^34-(10256432620616693/8)n^32+(96644962904826490/3)n^30-(55374779502235521605/78)n^28+13595007345878292201n^26-(1343025665002947503585/6)n^24+(103282404791891847624350/33)n^22-(293011473877695719531693/8)n^20+(4230578382216795982208675/12)n^18-(852579299017607597744223823/312)n^16+16612373793168429383959075n^14-(10109211367224080302067095289/132)n^12+(1536415892125531292757887885/6)n^10-(2335072151355459428197254585/4)n^8+(2484218879788644160052707565/3)n^6-(196329117914923619018719464145/312)n^4+(8415972489098354019612117925/44)n^2

S(n,52)=(1/53)n^53+(1/2)n^52+(13/3)n^51-(1105/6)n^49+(30940/3)n^47-(1672307/3)n^45+(83615350/

3)n^43-(3822250607/3)n^41+(158753389900/3)n^39-(11918497460779/6)n^37+(201062473783405/3)n^35-(133333624068017009/66)n^33+(162114131324225080/3)n^31-(3818950310499001490/3)n^29+(78548931331741243828/3)n^27-(6983733458015327018642/15)n^25+(233508045616451133759400/33)n^23-(544164165772863479130287/6)n^21+(2894606261516755145721725/3)n^19-(50151723471623976337895519/6)n^17+(172768687448951665593174380/3)n^15-(10109211367224080302067095289/33)n^13+(3631528472296710328336825910/3)n^11-(10118645989206990855521436535/3)n^9+(18454197392715642331

自然数幂和公式(1-267)

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