History of Touchard polynomials $T_n$

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when	who	what
2026-09-16 11:22	bmatschke interactive	tags: Sheffer sequence, polynomials of binomial type. a tag is a family of tables, and these are the members of one. numberdb-data#112 asked for it; the membership is decided by the generating function each table gives, not by its title. a tag is a family of tables, and these are the members of one.	current reviewed
2026-09-12 15:25	bmatschke	attached generate.py
2026-08-27 01:10	bmatschke with claude-opus-5 interactive	the link back, which was only going one way
2026-08-26 14:34	bmatschke generator	the Touchard polynomials, coefficients being the Stirling numbers of the second kind
2026-08-26 14:34	bmatschke with claude-opus-5 interactive	proposed from touchard.yaml

What changed between 2026-09-12 15:25 and 2026-09-16 11:22

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 - combinatorics - generating function+- Sheffer sequence+- polynomials of binomial type Programs:   program-sage:

from line 59 (65 lines, 50 fewer than before) @@ -57,115 +59,65 @@

       \ next one after this table" Numbers:-- params:-    n: '0'-  number: '1'-- params:-    n: '1'-  number: x-- params:-    n: '2'-  number: x^2 + x-- params:-    n: '3'-  number: x^3 + 3*x^2 + x-- params:-    n: '4'-  number: x^4 + 6*x^3 + 7*x^2 + x-- params:-    n: '5'-  number: x^5 + 10*x^4 + 25*x^3 + 15*x^2 + x-- params:-    n: '6'-  number: x^6 + 15*x^5 + 65*x^4 + 90*x^3 + 31*x^2 + x-- params:-    n: '7'-  number: x^7 + 21*x^6 + 140*x^5 + 350*x^4 + 301*x^3 + 63*x^2 + x-- params:-    n: '8'-  number: x^8 + 28*x^7 + 266*x^6 + 1050*x^5 + 1701*x^4 + 966*x^3 + 127*x^2 + x-- params:-    n: '9'-  number: x^9 + 36*x^8 + 462*x^7 + 2646*x^6 + 6951*x^5 + 7770*x^4 + 3025*x^3 + 255*x^2-    + x-- params:-    n: '10'-  number: x^10 + 45*x^9 + 750*x^8 + 5880*x^7 + 22827*x^6 + 42525*x^5 + 34105*x^4 +-    9330*x^3 + 511*x^2 + x-- params:-    n: '11'-  number: x^11 + 55*x^10 + 1155*x^9 + 11880*x^8 + 63987*x^7 + 179487*x^6 + 246730*x^5+  '0': '1'+  '1': x+  '2': x^2 + x+  '3': x^3 + 3*x^2 + x+  '4': x^4 + 6*x^3 + 7*x^2 + x+  '5': x^5 + 10*x^4 + 25*x^3 + 15*x^2 + x+  '6': x^6 + 15*x^5 + 65*x^4 + 90*x^3 + 31*x^2 + x+  '7': x^7 + 21*x^6 + 140*x^5 + 350*x^4 + 301*x^3 + 63*x^2 + x+  '8': x^8 + 28*x^7 + 266*x^6 + 1050*x^5 + 1701*x^4 + 966*x^3 + 127*x^2 + x+  '9': x^9 + 36*x^8 + 462*x^7 + 2646*x^6 + 6951*x^5 + 7770*x^4 + 3025*x^3 + 255*x^2+    + x+  '10': x^10 + 45*x^9 + 750*x^8 + 5880*x^7 + 22827*x^6 + 42525*x^5 + 34105*x^4 + 9330*x^3+    + 511*x^2 + x+  '11': x^11 + 55*x^10 + 1155*x^9 + 11880*x^8 + 63987*x^7 + 179487*x^6 + 246730*x^5     + 145750*x^4 + 28501*x^3 + 1023*x^2 + x-- params:-    n: '12'-  number: x^12 + 66*x^11 + 1705*x^10 + 22275*x^9 + 159027*x^8 + 627396*x^7 + 1323652*x^6+  '12': x^12 + 66*x^11 + 1705*x^10 + 22275*x^9 + 159027*x^8 + 627396*x^7 + 1323652*x^6     + 1379400*x^5 + 611501*x^4 + 86526*x^3 + 2047*x^2 + x-- params:-    n: '13'-  number: x^13 + 78*x^12 + 2431*x^11 + 39325*x^10 + 359502*x^9 + 1899612*x^8 + 5715424*x^7+  '13': x^13 + 78*x^12 + 2431*x^11 + 39325*x^10 + 359502*x^9 + 1899612*x^8 + 5715424*x^7     + 9321312*x^6 + 7508501*x^5 + 2532530*x^4 + 261625*x^3 + 4095*x^2 + x-- params:-    n: '14'-  number: x^14 + 91*x^13 + 3367*x^12 + 66066*x^11 + 752752*x^10 + 5135130*x^9 + 20912320*x^8+  '14': x^14 + 91*x^13 + 3367*x^12 + 66066*x^11 + 752752*x^10 + 5135130*x^9 + 20912320*x^8     + 49329280*x^7 + 63436373*x^6 + 40075035*x^5 + 10391745*x^4 + 788970*x^3 + 8191*x^2     + x-- params:-    n: '15'-  number: x^15 + 105*x^14 + 4550*x^13 + 106470*x^12 + 1479478*x^11 + 12662650*x^10-    + 67128490*x^9 + 216627840*x^8 + 408741333*x^7 + 420693273*x^6 + 210766920*x^5-    + 42355950*x^4 + 2375101*x^3 + 16383*x^2 + x-- params:-    n: '16'-  number: x^16 + 120*x^15 + 6020*x^14 + 165620*x^13 + 2757118*x^12 + 28936908*x^11-    + 193754990*x^10 + 820784250*x^9 + 2141764053*x^8 + 3281882604*x^7 + 2734926558*x^6+  '15': x^15 + 105*x^14 + 4550*x^13 + 106470*x^12 + 1479478*x^11 + 12662650*x^10 ++    67128490*x^9 + 216627840*x^8 + 408741333*x^7 + 420693273*x^6 + 210766920*x^5 ++    42355950*x^4 + 2375101*x^3 + 16383*x^2 + x+  '16': x^16 + 120*x^15 + 6020*x^14 + 165620*x^13 + 2757118*x^12 + 28936908*x^11 ++    193754990*x^10 + 820784250*x^9 + 2141764053*x^8 + 3281882604*x^7 + 2734926558*x^6     + 1096190550*x^5 + 171798901*x^4 + 7141686*x^3 + 32767*x^2 + x-- params:-    n: '17'-  number: x^17 + 136*x^16 + 7820*x^15 + 249900*x^14 + 4910178*x^13 + 62022324*x^12-    + 512060978*x^11 + 2758334150*x^10 + 9528822303*x^9 + 20415995028*x^8 + 25708104786*x^7+  '17': x^17 + 136*x^16 + 7820*x^15 + 249900*x^14 + 4910178*x^13 + 62022324*x^12 ++    512060978*x^11 + 2758334150*x^10 + 9528822303*x^9 + 20415995028*x^8 + 25708104786*x^7     + 17505749898*x^6 + 5652751651*x^5 + 694337290*x^4 + 21457825*x^3 + 65535*x^2     + x-- params:-    n: '18'-  number: x^18 + 153*x^17 + 9996*x^16 + 367200*x^15 + 8408778*x^14 + 125854638*x^13+  '18': x^18 + 153*x^17 + 9996*x^16 + 367200*x^15 + 8408778*x^14 + 125854638*x^13     + 1256328866*x^12 + 8391004908*x^11 + 37112163803*x^10 + 106175395755*x^9 + 189036065010*x^8     + 197462483400*x^7 + 110687251039*x^6 + 28958095545*x^5 + 2798806985*x^4 + 64439010*x^3     + 131071*x^2 + x-- params:-    n: '19'-  number: x^19 + 171*x^18 + 12597*x^17 + 527136*x^16 + 13916778*x^15 + 243577530*x^14+  '19': x^19 + 171*x^18 + 12597*x^17 + 527136*x^16 + 13916778*x^15 + 243577530*x^14     + 2892439160*x^13 + 23466951300*x^12 + 129413217791*x^11 + 477297033785*x^10 +     1144614626805*x^9 + 1709751003480*x^8 + 1492924634839*x^7 + 693081601779*x^6 +     147589284710*x^5 + 11259666950*x^4 + 193448101*x^3 + 262143*x^2 + x-- params:-    n: '20'-  number: x^20 + 190*x^19 + 15675*x^18 + 741285*x^17 + 22350954*x^16 + 452329200*x^15+  '20': x^20 + 190*x^19 + 15675*x^18 + 741285*x^17 + 22350954*x^16 + 452329200*x^15     + 6302524580*x^14 + 61068660380*x^13 + 411016633391*x^12 + 1900842429486*x^11     + 5917584964655*x^10 + 12011282644725*x^9 + 15170932662679*x^8 + 11143554045652*x^7     + 4306078895384*x^6 + 749206090500*x^5 + 45232115901*x^4 + 580606446*x^3 + 524287*x^2     + x-- params:-    n: '21'-  number: x^21 + 210*x^20 + 19285*x^19 + 1023435*x^18 + 34952799*x^17 + 809944464*x^16+  '21': x^21 + 210*x^20 + 19285*x^19 + 1023435*x^18 + 34952799*x^17 + 809944464*x^16     + 13087462580*x^15 + 149304004500*x^14 + 1204909218331*x^13 + 6833042030178*x^12     + 26826851689001*x^11 + 71187132291275*x^10 + 123272476465204*x^9 + 132511015347084*x^8     + 82310957214948*x^7 + 26585679462804*x^6 + 3791262568401*x^5 + 181509070050*x^4     + 1742343625*x^3 + 1048575*x^2 + x-- params:-    n: '22'-  number: x^22 + 231*x^21 + 23485*x^20 + 1389850*x^19 + 53374629*x^18 + 1404142047*x^17+  '22': x^22 + 231*x^21 + 23485*x^20 + 1389850*x^19 + 53374629*x^18 + 1404142047*x^17     + 26046574004*x^16 + 345615943200*x^15 + 3295165281331*x^14 + 22496861868481*x^13     + 108823356051137*x^12 + 366282500870286*x^11 + 835143799377954*x^10 + 1241963303533920*x^9     + 1142399079991620*x^8 + 602762379967440*x^7 + 163305339345225*x^6 + 19137821912055*x^5     + 727778623825*x^4 + 5228079450*x^3 + 2097151*x^2 + x-- params:-    n: '23'-  number: x^23 + 253*x^22 + 28336*x^21 + 1859550*x^20 + 79781779*x^19 + 2364885369*x^18+  '23': x^23 + 253*x^22 + 28336*x^21 + 1859550*x^20 + 79781779*x^19 + 2364885369*x^18     + 49916988803*x^17 + 762361127264*x^16 + 8479404429331*x^15 + 68629175807115*x^14     + 401282560341390*x^13 + 1672162773483930*x^12 + 4864251308951100*x^11 + 9593401297313460*x^10     + 12320068811796900*x^9 + 9741955019900400*x^8 + 4382641999117305*x^7 + 998969857983405*x^6     + 96416888184100*x^5 + 2916342574750*x^4 + 15686335501*x^3 + 4194303*x^2 + x-- params:-    n: '24'-  number: x^24 + 276*x^23 + 33902*x^22 + 2454606*x^21 + 116972779*x^20 + 3880739170*x^19+  '24': x^24 + 276*x^23 + 33902*x^22 + 2454606*x^21 + 116972779*x^20 + 3880739170*x^19     + 92484925445*x^18 + 1610949936915*x^17 + 20677182465555*x^16 + 195820242247080*x^15     + 1362091021641000*x^14 + 6888836057922000*x^13 + 24930204590758260*x^12 + 63100165695775560*x^11

from line 125 (5 lines, 2 fewer than before) @@ -173,7 +125,5 @@

     + 6090236036084530*x^6 + 485000783495250*x^5 + 11681056634501*x^4 + 47063200806*x^3     + 8388607*x^2 + x-- params:-    n: '25'-  number: x^25 + 300*x^24 + 40250*x^23 + 3200450*x^22 + 168519505*x^21 + 6220194750*x^20+  '25': x^25 + 300*x^24 + 40250*x^23 + 3200450*x^22 + 168519505*x^21 + 6220194750*x^20     + 166218969675*x^19 + 3275678594925*x^18 + 48063331393110*x^17 + 526655161695960*x^16     + 4299394655347200*x^15 + 25958110360896000*x^14 + 114485073343744260*x^13 + 362262620784874680*x^12

from line 131 (5 lines, 2 fewer than before) @@ -181,7 +131,5 @@

     + 690223721118368580*x^8 + 227832482998716310*x^7 + 37026417000002430*x^6 + 2436684974110751*x^5     + 46771289738810*x^4 + 141197991025*x^3 + 16777215*x^2 + x-- params:-    n: '26'-  number: x^26 + 325*x^25 + 47450*x^24 + 4126200*x^23 + 238929405*x^22 + 9759104355*x^21+  '26': x^26 + 325*x^25 + 47450*x^24 + 4126200*x^23 + 238929405*x^22 + 9759104355*x^21     + 290622864675*x^20 + 6433839018750*x^19 + 107025546101760*x^18 + 1343731795378830*x^17     + 12725877242482560*x^16 + 90449030191104000*x^15 + 477898618396288260*x^14 +

from line 138 (5 lines, 2 fewer than before) @@ -190,7 +138,5 @@

     + 1631853797991016600*x^7 + 224595186974125331*x^6 + 12230196160292565*x^5 + 187226356946265*x^4     + 423610750290*x^3 + 33554431*x^2 + x-- params:-    n: '27'-  number: x^27 + 351*x^26 + 55575*x^25 + 5265000*x^24 + 333832005*x^23 + 15015551265*x^22+  '27': x^27 + 351*x^26 + 55575*x^25 + 5265000*x^24 + 333832005*x^23 + 15015551265*x^22     + 495564056130*x^21 + 12246296312250*x^20 + 229268487458010*x^19 + 3270191625210510*x^18     + 35569317763922670*x^17 + 294063066070824960*x^16 + 1834634071262848260*x^15

from line 146 (5 lines, 2 fewer than before) @@ -200,7 +146,5 @@

     + 61338207158409090*x^5 + 749329038535350*x^4 + 1270865805301*x^3 + 67108863*x^2     + x-- params:-    n: '28'-  number: x^28 + 378*x^27 + 64701*x^26 + 6654375*x^25 + 460192005*x^24 + 22693687380*x^23+  '28': x^28 + 378*x^27 + 64701*x^26 + 6654375*x^25 + 460192005*x^24 + 22693687380*x^23     + 825906183960*x^22 + 22653141490980*x^21 + 474194413703010*x^20 + 7626292886912700*x^19     + 94432767017711850*x^18 + 898741468057510350*x^17 + 6539643128396047620*x^16

from line 154 (5 lines, 2 fewer than before) @@ -210,7 +154,5 @@

     + 8220146115188676396*x^6 + 307440364830580800*x^5 + 2998587019946701*x^4 + 3812664524766*x^3     + 134217727*x^2 + x-- params:-    n: '29'-  number: x^29 + 406*x^28 + 74907*x^27 + 8336601*x^26 + 626551380*x^25 + 33738295500*x^24+  '29': x^29 + 406*x^28 + 74907*x^27 + 8336601*x^26 + 626551380*x^25 + 33738295500*x^24     + 1347860993700*x^23 + 40823077538100*x^22 + 949910385013590*x^21 + 17110181160972900*x^20     + 239332331869053150*x^19 + 2598531274376323650*x^18 + 21818248085373723570*x^17

from line 162 (5 lines, 2 fewer than before) @@ -220,7 +162,5 @@

     + 588469772213874823272*x^7 + 49628317055962639176*x^6 + 1540200411172850701*x^5     + 11998160744311570*x^4 + 11438127792025*x^3 + 268435455*x^2 + x-- params:-    n: '30'-  number: x^30 + 435*x^29 + 86275*x^28 + 10359090*x^27 + 843303006*x^26 + 49402080000*x^25+  '30': x^30 + 435*x^29 + 86275*x^28 + 10359090*x^27 + 843303006*x^26 + 49402080000*x^25     + 2157580085700*x^24 + 71823880393200*x^23 + 1848018090851790*x^22 + 37058299246258290*x^21     + 581535955088511150*x^20 + 7145845579888333500*x^19 + 68591811024147549270*x^18

from line 171 (5 lines, 2 fewer than before) @@ -231,7 +171,5 @@

     + 7713000216608565075*x^5 + 48004081105038305*x^4 + 34314651811530*x^3 + 536870911*x^2     + x-- params:-    n: '31'-  number: x^31 + 465*x^30 + 98890*x^29 + 12774790*x^28 + 1122998436*x^27 + 71327958156*x^26+  '31': x^31 + 465*x^30 + 98890*x^29 + 12774790*x^28 + 1122998436*x^27 + 71327958156*x^26     + 3392632085700*x^25 + 123605802450000*x^24 + 3499967339895390*x^23 + 77714697244997670*x^22     + 1359760239259935240*x^21 + 18776564681658556500*x^20 + 204362877042025885770*x^19

from line 180 (5 lines, 2 fewer than before) @@ -242,7 +180,5 @@

     + 1803573616698300679617*x^6 + 38613005164147863680*x^5 + 192050639071964750*x^4     + 102944492305501*x^3 + 1073741823*x^2 + x-- params:-    n: '32'-  number: x^32 + 496*x^31 + 112840*x^30 + 15642600*x^29 + 1480692556*x^28 + 101648915928*x^27+  '32': x^32 + 496*x^31 + 112840*x^30 + 15642600*x^29 + 1480692556*x^28 + 101648915928*x^27     + 5247158997756*x^26 + 208421604592500*x^25 + 6466506598695390*x^24 + 158213946062591640*x^23     + 3069483578649883980*x^22 + 47331529706117196540*x^21 + 579894170675197015770*x^20

from line 189 (5 lines, 2 fewer than before) @@ -253,7 +189,5 @@

     + 208175663741028175181836*x^7 + 10860054705353951941382*x^6 + 193257076459811283150*x^5     + 768305500780164501*x^4 + 308834550658326*x^3 + 2147483647*x^2 + x-- params:-    n: '33'-  number: x^33 + 528*x^32 + 128216*x^31 + 19027800*x^30 + 1934327956*x^29 + 143108307496*x^28+  '33': x^33 + 528*x^32 + 128216*x^31 + 19027800*x^30 + 1934327956*x^29 + 143108307496*x^28     + 7991679727812*x^27 + 344847738534156*x^26 + 11677046713507890*x^25 + 313410104431281000*x^24     + 6708404338089491700*x^23 + 114860168436414644100*x^22 + 1573856294503658143110*x^21

from line 199 (5 lines, 2 fewer than before) @@ -265,7 +199,5 @@

     + 967053687799836580251*x^5 + 3073530837671316330*x^4 + 926505799458625*x^3 +     4294967295*x^2 + x-- params:-    n: '34'-  number: x^34 + 561*x^33 + 145112*x^32 + 23002496*x^31 + 2505161956*x^30 + 199203818220*x^29+  '34': x^34 + 561*x^33 + 145112*x^32 + 23002496*x^31 + 2505161956*x^30 + 199203818220*x^29     + 11998712337700*x^28 + 560623091185080*x^27 + 20643087915395946*x^26 + 605336272268978250*x^25     + 14230246844440235700*x^24 + 269153468212472953200*x^23 + 4100780000104780313310*x^22

from line 209 (5 lines, 2 fewer than before) @@ -277,7 +209,5 @@

     + 393088565539300974168903*x^6 + 4838341969836854217585*x^5 + 12295049856484723945*x^4     + 2779521693343170*x^3 + 8589934591*x^2 + x-- params:-    n: '35'-  number: x^35 + 595*x^34 + 163625*x^33 + 27646080*x^32 + 3218239332*x^31 + 274358676900*x^30+  '35': x^35 + 595*x^34 + 163625*x^33 + 27646080*x^32 + 3218239332*x^31 + 274358676900*x^30     + 17775623066080*x^29 + 896587036640680*x^28 + 35779911377393106*x^27 + 1142056558069272846*x^26     + 29363653651164691950*x^25 + 610679392479038610000*x^24 + 10291309768991658236910*x^23

from line 219 (5 lines, 2 fewer than before) @@ -289,7 +219,5 @@

     + 72786959006434393367186463*x^7 + 2363369735205642699231003*x^6 + 24204004899040755811870*x^5     + 49182978947632238950*x^4 + 8338573669964101*x^3 + 17179869183*x^2 + x-- params:-    n: '36'-  number: x^36 + 630*x^35 + 183855*x^34 + 33045705*x^33 + 4102913892*x^32 + 374124096192*x^31+  '36': x^36 + 630*x^35 + 183855*x^34 + 33045705*x^33 + 4102913892*x^32 + 374124096192*x^31     + 26006383373080*x^30 + 1412080105557000*x^29 + 60884348403332146*x^28 + 2108114165258886708*x^27     + 59057124160965785946*x^26 + 1344770733758155908750*x^25 + 24947615188488584876910*x^24

from line 230 (5 lines, 2 fewer than before) @@ -302,7 +230,5 @@

     + 14204422416132896951197888*x^6 + 121069207474151411298300*x^5 + 196740254364198919901*x^4     + 25015738189761486*x^3 + 34359738367*x^2 + x-- params:-    n: '37'-  number: x^37 + 666*x^36 + 205905*x^35 + 39296775*x^34 + 5193422157*x^33 + 505417340736*x^32+  '37': x^37 + 666*x^36 + 205905*x^35 + 39296775*x^34 + 5193422157*x^33 + 505417340736*x^32     + 37604230355032*x^31 + 2192271606749400*x^30 + 101834671464485146*x^29 + 3812875920552186796*x^28     + 115976206622955727062*x^27 + 2880255961943266343346*x^26 + 58566883532442482595660*x^25

from line 243 (5 lines, 2 fewer than before) @@ -317,7 +243,5 @@

     + 605542777625121255411401*x^5 + 786986033194985441090*x^4 + 75047248929022825*x^3     + 68719476735*x^2 + x-- params:-    n: '38'-  number: x^38 + 703*x^37 + 229881*x^36 + 46503450*x^35 + 6529512507*x^34 + 676800271917*x^33+  '38': x^38 + 703*x^37 + 229881*x^36 + 46503450*x^35 + 6529512507*x^34 + 676800271917*x^33     + 53777585258584*x^32 + 3358002747755392*x^31 + 167602819666967146*x^30 + 6766081393022256030*x^29     + 222736732398416957350*x^28 + 6011613540763070974020*x^27 + 133453538542967407522656*x^26

from line 256 (5 lines, 2 fewer than before) @@ -332,7 +256,5 @@

     + 512691165003254319966325169*x^6 + 3028500874158801262498095*x^5 + 3148019180028870787185*x^4     + 225141815506545210*x^3 + 137438953471*x^2 + x-- params:-    n: '39'-  number: x^39 + 741*x^38 + 255892*x^37 + 54779166*x^36 + 8157133257*x^35 + 898803697155*x^34+  '39': x^39 + 741*x^38 + 255892*x^37 + 54779166*x^36 + 8157133257*x^35 + 898803697155*x^34     + 76111994231845*x^33 + 5078885476030080*x^32 + 271700904847384298*x^31 + 11794165983031270410*x^30     + 418953092796062382220*x^29 + 12248242047918745779820*x^28 + 295767104143570323821196*x^27

from line 270 (5 lines, 2 fewer than before) @@ -348,7 +270,5 @@

     + 15145652389974035183277660*x^5 + 12592301861930989693950*x^4 + 675425583958589101*x^3     + 274877906943*x^2 + x-- params:-    n: '40'-  number: x^40 + 780*x^39 + 284050*x^38 + 64247170*x^37 + 10129183233*x^36 + 1184303361150*x^35+  '40': x^40 + 780*x^39 + 284050*x^38 + 64247170*x^37 + 10129183233*x^36 + 1184303361150*x^35     + 106671319935115*x^34 + 7590581285680965*x^33 + 434225240080346858*x^32 + 20216894033300183648*x^31     + 772778072287000494520*x^30 + 24397881739004554864200*x^29 + 638717881485295205656156*x^28