History of Bernoulli polynomials of the second kind $\psi_n$

 - number theory - generating function+- Sheffer sequence Programs:   program-sage:

       series[26] * factorial(26)       # the next one after this table' Numbers:-- params:-    n: '0'-  number: '1'-- params:-    n: '1'-  number: x + 1/2-- params:-    n: '2'-  number: x^2 - 1/6-- params:-    n: '3'-  number: x^3 - 3/2*x^2 + 1/4-- params:-    n: '4'-  number: x^4 - 4*x^3 + 4*x^2 - 19/30-- params:-    n: '5'-  number: x^5 - 15/2*x^4 + 55/3*x^3 - 15*x^2 + 9/4-- params:-    n: '6'-  number: x^6 - 12*x^5 + 105/2*x^4 - 100*x^3 + 72*x^2 - 863/84-- params:-    n: '7'-  number: x^7 - 35/2*x^6 + 119*x^5 - 1575/4*x^4 + 1918/3*x^3 - 420*x^2 + 1375/24-- params:-    n: '8'-  number: x^8 - 24*x^7 + 700/3*x^6 - 1176*x^5 + 3248*x^4 - 4704*x^3 + 2880*x^2 - 33953/90-- params:-    n: '9'-  number: x^9 - 63/2*x^8 + 414*x^7 - 2940*x^6 + 60921/5*x^5 - 29547*x^4 + 39204*x^3-    - 22680*x^2 + 57281/20-- params:-    n: '10'-  number: x^10 - 40*x^9 + 1365/2*x^8 - 6480*x^7 + 37415*x^6 - 134568*x^5 + 295310*x^4+  '0': '1'+  '1': x + 1/2+  '2': x^2 - 1/6+  '3': x^3 - 3/2*x^2 + 1/4+  '4': x^4 - 4*x^3 + 4*x^2 - 19/30+  '5': x^5 - 15/2*x^4 + 55/3*x^3 - 15*x^2 + 9/4+  '6': x^6 - 12*x^5 + 105/2*x^4 - 100*x^3 + 72*x^2 - 863/84+  '7': x^7 - 35/2*x^6 + 119*x^5 - 1575/4*x^4 + 1918/3*x^3 - 420*x^2 + 1375/24+  '8': x^8 - 24*x^7 + 700/3*x^6 - 1176*x^5 + 3248*x^4 - 4704*x^3 + 2880*x^2 - 33953/90+  '9': x^9 - 63/2*x^8 + 414*x^7 - 2940*x^6 + 60921/5*x^5 - 29547*x^4 + 39204*x^3 -+    22680*x^2 + 57281/20+  '10': x^10 - 40*x^9 + 1365/2*x^8 - 6480*x^7 + 37415*x^6 - 134568*x^5 + 295310*x^4     - 365280*x^3 + 201600*x^2 - 3250433/132-- params:-    n: '11'-  number: x^11 - 99/2*x^10 + 3190/3*x^9 - 51975/4*x^8 + 99429*x^7 - 987525/2*x^6 +-    1592096*x^5 - 3224925*x^4 + 3764112*x^3 - 1995840*x^2 + 1891755/8-- params:-    n: '12'-  number: x^12 - 60*x^11 + 1584*x^10 - 24200*x^9 + 473319/2*x^8 - 1546380*x^7 + 6833860*x^6+  '11': x^11 - 99/2*x^10 + 3190/3*x^9 - 51975/4*x^8 + 99429*x^7 - 987525/2*x^6 + 1592096*x^5+    - 3224925*x^4 + 3764112*x^3 - 1995840*x^2 + 1891755/8+  '12': x^12 - 60*x^11 + 1584*x^10 - 24200*x^9 + 473319/2*x^8 - 1546380*x^7 + 6833860*x^6     - 20182800*x^5 + 38260728*x^4 - 42514560*x^3 + 21772800*x^2 - 13695779093/5460-- params:-    n: '13'-  number: x^13 - 143/2*x^12 + 2275*x^11 - 42471*x^10 + 1548833/3*x^9 - 17144127/4*x^8+  '13': x^13 - 143/2*x^12 + 2275*x^11 - 42471*x^10 + 1548833/3*x^9 - 17144127/4*x^8     + 173413955/7*x^7 - 99657415*x^6 + 1368354988/5*x^5 - 490483422*x^4 + 522356640*x^3     - 259459200*x^2 + 24466579093/840-- params:-    n: '14'-  number: x^14 - 84*x^13 + 19019/6*x^12 - 70980*x^11 + 5246241/5*x^10 - 10774764*x^9+  '14': x^14 - 84*x^13 + 19019/6*x^12 - 70980*x^11 + 5246241/5*x^10 - 10774764*x^9     + 314931617/4*x^8 - 412140300*x^7 + 4600447852/3*x^6 - 19796208432/5*x^5 + 6760458432*x^4     - 6936733440*x^3 + 3353011200*x^2 - 132282840127/360-- params:-    n: '15'-  number: x^15 - 195/2*x^14 + 4305*x^13 - 455455/4*x^12 + 2010645*x^11 - 50008959/2*x^10+  '15': x^15 - 195/2*x^14 + 4305*x^13 - 455455/4*x^12 + 2010645*x^11 - 50008959/2*x^10     + 675182365/3*x^9 - 11864147295/8*x^8 + 7148825970*x^7 - 24894259390*x^6 + 60941259288*x^5     - 99737688960*x^4 + 99013795200*x^3 - 46702656000*x^2 + 240208245823/48-- params:-    n: '16'-  number: x^16 - 112*x^15 + 5720*x^14 - 176400*x^13 + 10998988/3*x^12 - 54272400*x^11+  '16': x^16 - 112*x^15 + 5720*x^14 - 176400*x^13 + 10998988/3*x^12 - 54272400*x^11     + 589458584*x^10 - 14301086800/3*x^9 + 28818645856*x^8 - 129516266880*x^7 + 1277772845440/3*x^6     - 995165633280*x^5 + 1568627191296*x^4 - 1511816785920*x^3 + 697426329600*x^2     - 111956703448001/1530-- params:-    n: '17'-  number: x^17 - 255/2*x^16 + 22372/3*x^15 - 265200*x^14 + 6407198*x^13 - 111291180*x^12+  '17': x^17 - 255/2*x^16 + 22372/3*x^15 - 265200*x^14 + 6407198*x^13 - 111291180*x^12     + 1434329780*x^11 - 13952967600*x^10 + 928729202401/9*x^9 - 579706822695*x^8 +     2452060831480*x^7 - 7669304479200*x^6 + 85968926965008/5*x^5 - 26204724862560*x^4     + 24588590342400*x^3 - 11115232128000*x^2 + 4573423873125/4-- params:-    n: '18'-  number: x^18 - 144*x^17 + 19125/2*x^16 - 388416*x^15 + 10792314*x^14 - 217318752*x^13+  '18': x^18 - 144*x^17 + 19125/2*x^16 - 388416*x^15 + 10792314*x^14 - 217318752*x^13     + 3277547130*x^12 - 37729897920*x^11 + 1673578597977/5*x^10 - 2293802567056*x^9     + 12092677825410*x^8 - 48501172437120*x^7 + 145098027700272*x^6 - 1567399479766272/5*x^5     + 463465101767040*x^4 - 424405694361600*x^3 + 188305108992000*x^2 - 30342376302478019/1596-- params:-    n: '19'-  number: x^19 - 323/2*x^18 + 12084*x^17 - 2223855/4*x^16 + 88011686/5*x^15 - 406668951*x^14+  '19': x^19 - 323/2*x^18 + 12084*x^17 - 2223855/4*x^16 + 88011686/5*x^15 - 406668951*x^14     + 7093171732*x^13 - 95320932135*x^12 + 998233909257*x^11 - 81854000736651/10*x^10     + 157521686886928/3*x^9 - 261792607187475*x^8 + 7011240335453296/7*x^7 - 2879449701839688*x^6     + 30082965538822272/5*x^5 - 8652611611967040*x^4 + 7748235407001600*x^3 - 3379030566912000*x^2     + 56310194579604163/168-- params:-    n: '20'-  number: x^20 - 180*x^19 + 45220/3*x^18 - 779760*x^17 + 55809555/2*x^16 - 733052376*x^15+  '20': x^20 - 180*x^19 + 45220/3*x^18 - 779760*x^17 + 55809555/2*x^16 - 733052376*x^15     + 102469372720/7*x^14 - 227016467280*x^13 + 2769288977455*x^12 - 26746824379860*x^11     + 204835481465316*x^10 - 1239825958993440*x^9 + 5882812601374960*x^8 - 151030551841260480/7*x^7     + 59835707603071680*x^6 - 121285016030877696*x^5 + 170061247969113600*x^4 - 149179920390144000*x^3     + 64023737057280000*x^2 - 12365722323469980029/1980-- params:-    n: '21'-  number: x^21 - 399/2*x^20 + 18585*x^19 - 1073975*x^18 + 43132698*x^17 - 5113194975/4*x^16+  '21': x^21 - 399/2*x^20 + 18585*x^19 - 1073975*x^18 + 43132698*x^17 - 5113194975/4*x^16     + 28970107082*x^15 - 513378767850*x^14 + 7213058608077*x^13 - 161982267131685/2*x^12     + 8020363165546095/11*x^11 - 5258103386481855*x^10 + 90675458929607272/3*x^9 -

     - 3510201084291054720*x^4 + 3020956027723468800*x^3 - 1277273554292736000*x^2     + 161867055619224199787/1320-- params:-    n: '22'-  number: x^22 - 220*x^21 + 45353/2*x^20 - 1455300*x^19 + 195535802/3*x^18 - 2164128120*x^17+  '22': x^22 - 220*x^21 + 45353/2*x^20 - 1455300*x^19 + 195535802/3*x^18 - 2164128120*x^17     + 220944743965/4*x^16 - 1108963070600*x^15 + 17773292421313*x^14 - 229451847983820*x^13     + 4794295038648115/2*x^12 - 20284599731022900*x^11 + 693338933092243856/5*x^10

     + 29471976683031856512*x^6 - 56632097478664350720*x^5 + 75920678645023872000*x^4     - 64188285602918400000*x^3 + 26761922089943040000*x^2 - 20953816286242674495191/8280-- params:-    n: '23'-  number: x^23 - 483/2*x^22 + 82225/3*x^21 - 7772919/4*x^20 + 96505332*x^19 - 3567769821*x^18+  '23': x^23 - 483/2*x^22 + 82225/3*x^21 - 7772919/4*x^20 + 96505332*x^19 - 3567769821*x^18     + 101862493150*x^17 - 18396761470395/8*x^16 + 625338073007263/15*x^15 - 1225903285065459/2*x^14     + 7350842198684775*x^13 - 288270768999258045/4*x^12 + 577130864847801178*x^11

     - 1717133441463494208000*x^4 + 1427876882647971840000*x^3 - 587545834974658560000*x^2     + 4380881778942163832799/80-- params:-    n: '24'-  number: x^24 - 264*x^23 + 32844*x^22 - 2560360*x^21 + 701379756/5*x^20 - 5742375408*x^19+  '24': x^24 - 264*x^23 + 32844*x^22 - 2560360*x^21 + 701379756/5*x^20 - 5742375408*x^19     + 546869431768/3*x^18 - 4596834498960*x^17 + 187147249265823/2*x^16 - 7770003687519304/5*x^15     + 21193792880719212*x^14 - 238166141203073160*x^13 + 2206461762371899472*x^12

     - 31433687292974729748480*x^5 + 40536880042625584128000*x^4 - 33187814234683637760000*x^3     + 13488008733331292160000*x^2 - 101543126947618093900697699/81900-- params:-    n: '25'-  number: x^25 - 575/2*x^24 + 39050*x^23 - 3332700*x^22 + 601513825/3*x^21 - 9043336995*x^20+  '25': x^25 - 575/2*x^24 + 39050*x^23 - 3332700*x^22 + 601513825/3*x^21 - 9043336995*x^20     + 317469004600*x^19 - 26669126400400/3*x^18 + 201871340883575*x^17 - 15037787741528475/4*x^16     + 173509032243521030/3*x^15 - 738136989823775100*x^14 + 101759601425189238025/13*x^13