History of Stirling polynomials $S_k(x)$

 Display properties:   number-header: $S_k(x)$-Numbers: []+Numbers:+- params:+    k: '0'+  number: '1'+- params:+    k: '1'+  number: 1/2*x + 1/2+- params:+    k: '2'+  number: 1/4*x^2 + 5/12*x + 1/6+- params:+    k: '3'+  number: 1/8*x^3 + 1/4*x^2 + 1/8*x+- params:+    k: '4'+  number: 1/16*x^4 + 1/8*x^3 + 1/48*x^2 - 3/40*x - 1/30+- params:+    k: '5'+  number: 1/32*x^5 + 5/96*x^4 - 5/96*x^3 - 13/96*x^2 - 1/16*x+- params:+    k: '6'+  number: 1/64*x^6 + 1/64*x^5 - 5/64*x^4 - 77/576*x^3 - 1/48*x^2 + 59/1008*x + 1/42+- params:+    k: '7'+  number: 1/128*x^7 - 7/96*x^5 - 49/576*x^4 + 91/1152*x^3 + 97/576*x^2 + 5/72*x+- params:+    k: '8'+  number: 1/256*x^8 - 1/192*x^7 - 7/128*x^6 - 7/288*x^5 + 1211/6912*x^4 + 407/1728*x^3+    + 221/8640*x^2 - 193/2160*x - 1/30+- params:+    k: '9'+  number: 1/512*x^9 - 3/512*x^8 - 9/256*x^7 + 91/3840*x^6 + 1687/7680*x^5 + 93/512*x^4+    - 197/960*x^3 - 671/1920*x^2 - 21/160*x+- params:+    k: '10'+  number: 1/1024*x^10 - 5/1024*x^9 - 5/256*x^8 + 77/1536*x^7 + 623/3072*x^6 + 115/9216*x^5+    - 863/1536*x^4 - 1421/2304*x^3 - 5/128*x^2 + 691/3168*x + 5/66+- params:+    k: '11'+  number: 1/2048*x^11 - 11/3072*x^10 - 55/6144*x^9 + 11/192*x^8 + 2651/18432*x^7 -+    1837/9216*x^6 - 5225/6144*x^5 - 2365/4608*x^4 + 3487/4608*x^3 + 413/384*x^2 ++    3/8*x+- params:+    k: '12'+  number: 1/4096*x^12 - 5/2048*x^11 - 11/4096*x^10 + 319/6144*x^9 + 869/12288*x^8+    - 759/2048*x^7 - 98219/110592*x^6 + 4235/18432*x^5 + 66715/27648*x^4 + 51977/23040*x^3+    + 3149/60480*x^2 - 11247/14560*x - 691/2730+- params:+    k: '13'+  number: 1/8192*x^13 - 13/8192*x^12 + 13/24576*x^11 + 1001/24576*x^10 + 143/24576*x^9+    - 232661/516096*x^8 - 4795219/7741440*x^7 + 1657799/1105920*x^6 + 2318173/552960*x^5+    + 513851/276480*x^4 - 3596333/967680*x^3 - 1110689/241920*x^2 - 7601/5040*x+- params:+    k: '14'+  number: 1/16384*x^14 - 49/49152*x^13 + 91/49152*x^12 + 7007/245760*x^11 - 29029/737280*x^10+    - 21307/49152*x^9 - 289003/2211840*x^8 + 18737719/6635520*x^7 + 7794787/1658880*x^6+    - 2241967/829440*x^5 - 5567081/414720*x^4 - 4576061/414720*x^3 + 14827/103680*x^2+    + 16139/4320*x + 7/6+- params:+    k: '15'+  number: 1/32768*x^15 - 5/8192*x^14 + 35/16384*x^13 + 2639/147456*x^12 - 1547/24576*x^11+    - 50765/147456*x^10 + 183469/442368*x^9 + 1611467/442368*x^8 + 2450305/884736*x^7+    - 5430607/442368*x^6 - 5744219/221184*x^5 - 913789/110592*x^4 + 648949/27648*x^3+    + 29921/1152*x^2 + 65/8*x+- params:+    k: '16'+  number: 1/65536*x^16 - 3/8192*x^15 + 95/49152*x^14 + 91/9216*x^13 - 20293/294912*x^12+    - 2717/12288*x^11 + 380237/442368*x^10 + 99385/27648*x^9 - 9007999/5308416*x^8+    - 5339789/221184*x^7 - 736853/24576*x^6 + 769285/27648*x^5 + 52181287/552960*x^4+    + 321517/4608*x^3 - 66293/20736*x^2 - 347341/14688*x - 3617/510+- params:+    k: '17'+  number: 1/131072*x^17 - 85/393216*x^16 + 51/32768*x^15 + 1309/294912*x^14 - 111503/1769472*x^13+    - 173927/1769472*x^12 + 108953/98304*x^11 + 7062055/2654208*x^10 - 80283775/10616832*x^9+    - 1729577603/53084160*x^8 - 17489719/1658880*x^7 + 766204229/6635520*x^6 + 660607369/3317760*x^5+    + 140907917/3317760*x^4 - 8562679/46080*x^3 - 4336337/23040*x^2 - 3617/64*x+- params:+    k: '18'+  number: 1/262144*x^18 - 33/262144*x^17 + 153/131072*x^16 + 221/196608*x^15 - 20179/393216*x^14+    + 221/131072*x^13 + 841789/737280*x^12 + 3313453/2949120*x^11 - 152951227/11796480*x^10+    - 10085915567/318504960*x^9 + 714638197/17694720*x^8 + 3162255493/13271040*x^7+    + 10505099/46080*x^6 - 410886073/1327104*x^5 - 302682773/368640*x^4 - 2417398637/4354560*x^3+    + 20498021/483840*x^2 + 218052091/1149120*x + 43867/798+- params:+    k: '19'+  number: 1/524288*x^19 - 19/262144*x^18 + 437/524288*x^17 - 323/491520*x^16 - 149549/3932160*x^15+    + 81719/1179648*x^14 + 11793053/11796480*x^13 - 1734187/2949120*x^12 - 1143551461/70778880*x^11+    - 5876093197/318504960*x^10 + 513442731217/4459069440*x^9 + 61504274653/185794560*x^8+    - 3104327483/185794560*x^7 - 1680599039/1327104*x^6 - 24689670707/13271040*x^5+    - 1563405709/6967296*x^4 + 4494370181/2488320*x^3 + 19594583/11520*x^2 + 745739/1512*x+- params:+    k: '20'+  number: 1/1048576*x^20 - 65/1572864*x^19 + 1805/3145728*x^18 - 2261/1572864*x^17+    - 121771/4718592*x^16 + 1732895/16515072*x^15 + 24825457/33030144*x^14 - 14641913/7077888*x^13+    - 152528675/9437184*x^12 + 680863651/127401984*x^11 + 1015992437629/5350883328*x^10+    + 801828182339/2675441664*x^9 - 39208296545/55738368*x^8 - 43572202903/15925248*x^7+    - 32314359611/15925248*x^6 + 5444607417793/1393459200*x^5 + 1815290979443/209018880*x^4+    + 283991612543/52254720*x^3 - 607681817/1064448*x^2 - 18784524793/9979200*x -+    174611/330+- params:+    k: '21'+  number: 1/2097152*x^21 - 49/2097152*x^20 + 805/2097152*x^19 - 10241/6291456*x^18+    - 6251/393216*x^17 + 120479/1048576*x^16 + 13273039/28311552*x^15 - 3199315/1048576*x^14+    - 738354421/56623104*x^13 + 190047936715/5605687296*x^12 + 1308403256225/5605687296*x^11+    + 35993435555/509607936*x^10 - 762535696813/424673280*x^9 - 27215774249/7077888*x^8+    + 4250008483/2654208*x^7 + 12910570050283/796262400*x^6 + 8311577861809/398131200*x^5+    + 3665539381/4976640*x^4 - 128833133063/6082560*x^3 - 71328391847/3801600*x^2+    - 23223263/4400*x+- params:+    k: '22'+  number: 1/4194304*x^22 - 55/4194304*x^21 + 3157/12582912*x^20 - 19019/12582912*x^19+    - 54131/6291456*x^18 + 2028763/18874368*x^17 + 11728453/56623104*x^16 - 194970875/56623104*x^15+    - 871710785/113246208*x^14 + 303923002747/5096079360*x^13 + 1124746504037/5096079360*x^12+    - 5488479812849/15288238080*x^11 - 1249125008747/424673280*x^10 - 11331369254557/3822059520*x^9+    + 1305289773953/106168320*x^8 + 58136290940783/1592524800*x^7 + 328916159561/15925248*x^6+    - 67624515746711/1194393600*x^5 - 15198170263/138240*x^4 - 4796904302521/74649600*x^3+    + 19609193/2304*x^2 + 67430533733/2980800*x + 854513/138+- params:+    k: '23'+  number: 1/8388608*x^23 - 23/3145728*x^22 + 253/1572864*x^21 - 5313/4194304*x^20+    - 278047/75497472*x^19 + 379753/4194304*x^18 - 4807/35389440*x^17 - 2805494989/849346560*x^16+    - 5007004849/3397386240*x^15 + 193484597929/2548039680*x^14 + 549689116147/3822059520*x^13+    - 13765722903649/15288238080*x^12 - 108131369737513/30576476160*x^11 + 533288632877/339738624*x^10+    + 12896761584443/424673280*x^9 + 54068692472531/1061683200*x^8 - 395036620048477/9555148800*x^7+    - 1153019519149003/4777574400*x^6 - 33156550213997/119439360*x^5 + 1264406584061/99532800*x^4+    + 612140294039/2073600*x^3 + 42884916121/172800*x^2 + 543781/8*x+- params:+    k: '24'+  number: 1/16777216*x^24 - 17/4194304*x^23 + 851/8388608*x^22 - 93863/94371840*x^21+    - 54901/83886080*x^20 + 2648657/37748736*x^19 - 39854837/283115520*x^18 - 24597419/8847360*x^17+    + 9778740697/2264924160*x^16 + 403350623743/5096079360*x^15 + 156626119601/10192158720*x^14+    - 3556808992681/2548039680*x^13 - 543457191427337/183458856960*x^12 + 157236667798331/15288238080*x^11+    + 2280762988276091/45864714240*x^10 + 530276656409269/19110297600*x^9 - 4424655011910617/19110297600*x^8+    - 896888976127001/1592524800*x^7 - 16667720756480383/71663616000*x^6 + 5629841337175471/5971968000*x^5+    + 10303015964096693/6270566400*x^4 + 472027022290043/522547200*x^3 - 14505391013281/101088000*x^2+    - 19098819511351/58968000*x - 236364091/2730+- params:+    k: '25'+  number: 1/33554432*x^25 - 75/33554432*x^24 + 3175/50331648*x^23 - 112585/150994944*x^22+    + 302335/301989888*x^21 + 15224275/301989888*x^20 - 49396985/226492416*x^19 -+    232634765/113246208*x^18 + 23685170575/2717908992*x^17 + 564960331985/8153726976*x^16+    - 20550211235/150994944*x^15 - 266263559870185/158997676032*x^14 - 850609190721865/953986056192*x^13+    + 1586065506783335/73383542784*x^12 + 524594688243635/9172942848*x^11 - 46307834639785279/642105999360*x^10+    - 3043109925477985/5350883328*x^9 - 2722928693387705/3567255552*x^8 + 19520336549170627/20065812480*x^7+    + 23790570574916621/5733089280*x^6 + 8672163370983461/2006581248*x^5 - 270154156186657/501645312*x^4+    - 4372254451065077/905748480*x^3 - 439060121079173/113218560*x^2 - 27181870465/26208*x+- params:+    k: '26'+  number: 1/67108864*x^26 - 247/201326592*x^25 + 325/8388608*x^24 - 162955/301989888*x^23+    + 3155945/1811939328*x^22 + 60698495/1811939328*x^21 - 221333255/905969664*x^20+    - 54679625/42467328*x^19 + 61144006495/5435817984*x^18 + 805959692135/16307453952*x^17+    - 23627873545055/85614133248*x^16 - 92202754097075/57076088832*x^15 + 2564377269108233/1027369598976*x^14+    + 14016717238133155/440301256704*x^13 + 8059177780550195/220150628352*x^12 - 1032090099442865177/3852635996160*x^11+    - 255383021614939933/275188285440*x^10 - 1319879630259619/9172942848*x^9 + 257419138953129023/53508833280*x^8+    + 343337398712800219/34398535680*x^7 + 329057696448850889/120394874880*x^6 - 77266476376935599/4299816960*x^5+    - 215564328553850663/7524679680*x^4 - 1037882972607731/69672960*x^3 + 71808945994969/26127360*x^2+    + 988411516871/181440*x + 8553103/6+- params:+    k: '27'+  number: 1/134217728*x^27 - 45/67108864*x^26 + 3159/134217728*x^25 - 3185/8388608*x^24+    + 258245/134217728*x^23 + 9521655/469762048*x^22 - 661763245/2818572288*x^21 -+    60270925/100663296*x^20 + 4798404325/402653184*x^19 + 132327470509/5435817984*x^18+    - 9494392741223/25367150592*x^17 - 3772139969191/3170893824*x^16 + 1484019759482027/228304355328*x^15+    + 589455699176021/16307453952*x^14 - 250415945707211/10871635968*x^13 - 150351100452672431/285380444160*x^12+    - 56183889487802063/57076088832*x^11 + 54350694833907043/23781703680*x^10 + 847310107567473563/71345111040*x^9+    + 228419594098511647/17836277760*x^8 - 141343896663787207/5945425920*x^7 - 364955742845955859/4459069440*x^6+    - 86878885394938817/1114767360*x^5 + 489404764452839/30965760*x^4 + 354454661873917/3870720*x^3+    + 380005230847/5376*x^2 + 148034475/8*x+- params:+    k: '28'+  number: 1/268435456*x^28 - 49/134217728*x^27 + 3801/268435456*x^26 - 35035/134217728*x^25+    + 483665/268435456*x^24 + 4304105/402653184*x^23 - 164441329/805306368*x^22 -+    22263241/402653184*x^21 + 26707071391/2415919104*x^20 - 10080497063/10871635968*x^19+    - 8984671231237/21743271936*x^18 - 5099856153497/10871635968*x^17 + 658631448480689/65229815808*x^16+    + 994591701251675/32614907904*x^15 - 23341301381804497/195689447424*x^14 - 122108104539133271/163074539520*x^13+    - 68565034338614333/244611809280*x^12 + 96938625562585807/13589544960*x^11 + 1171980497237001829/61152952320*x^10+    - 585162297969197/125829120*x^9 - 1699660549979226523/15288238080*x^8 - 2571052349102616353/12740198400*x^7+    - 137218691151170729/4777574400*x^6 + 51764291075542489/132710400*x^5 + 6887499125813227/11943936*x^4+    + 14177078267471623/49766400*x^3 - 123808606924159/2073600*x^2 - 133246171577173/1252800*x+    - 23749461029/870+- params:+    k: '29'+  number: 1/536870912*x^29 - 319/1610612736*x^28 + 13601/1610612736*x^27 - 472381/2684354560*x^26+    + 12411217/8053063680*x^25 + 6855745/1610612736*x^24 - 264730531/1610612736*x^23+    + 1547365963/4831838208*x^22 + 14763738847/1610612736*x^21 - 970368462427/43486543872*x^20+    - 50883554918261/130459631616*x^19 + 52066688758879/130459631616*x^18 + 1600161762948287/130459631616*x^17+    + 5228758068446641/391378894848*x^16 - 90886975025492851/391378894848*x^15 - 512039704655155057/652298158080*x^14+    + 1517359254562987639/978447237120*x^13 + 6642567038745999907/489223618560*x^12+    + 2398392605914874543/135895449600*x^11 - 42682015710304034779/611529523200*x^10+    - 16930980189568270943/61152952320*x^9 - 4026375753907450463/16986931200*x^8 ++    47930935750623964819/76441190400*x^7 + 1405538093353033667/764411904*x^6 + 513510281359061753/318504960*x^5+    - 60238334396942669/132710400*x^4 - 11034511862238329/5529600*x^3 - 57086114789377/38400*x^2+    - 30535021323/80*x+- params:+    k: '30'+  number: 1/1073741824*x^30 - 115/1073741824*x^29 + 5365/1073741824*x^28 - 125251/1073741824*x^27+    + 1333913/1073741824*x^26 + 311025/1073741824*x^25 - 1194537695/9663676416*x^24+    + 1732509155/3221225472*x^23 + 65526963775/9663676416*x^22 - 3246137774425/86973087744*x^21+    - 27412668787985/86973087744*x^20 + 107721586519655/86973087744*x^19 + 3211408209010345/260919263232*x^18+    - 3467958295874975/260919263232*x^17 - 28513955239770205/86973087744*x^16 - 5783755580597623139/11741366845440*x^15+    + 876534608478334451/195689447424*x^14 + 2723101105011846533/146767085568*x^13+    - 208999555459011863/30576476160*x^12 - 14815727048196555923/73383542784*x^11+    - 74232814489811081/169869312*x^10 + 13372048559724564593/45864714240*x^9 + 43755037609407055757/15288238080*x^8+    + 212033733867511888169/45864714240*x^7 + 251978613909759829/3822059520*x^6 -+    192330079560951796171/20065812480*x^5 - 7400959566028067593/557383680*x^4 - 479338827669299401/76640256*x^3+    + 5199005863541263/3548160*x^2 + 894582746652703/374976*x + 8615841276005/14322