History of Bateman polynomials $F_n(x)$

     CITE{Wiki}.   formula-orthogonality: If $B_n(x)=i^nF_n(ix)$, then $\int_{-\infty}^{\infty}B_m(x)B_n(x)\operatorname{sech}^2(\pi-    x/2)\,dx =\frac{4}{\pi(2n+1)}\delta_{m,n}$ CITE{Koelink}.+    x/2)\,\mathrm{d}x =\frac{4}{\pi(2n+1)}\delta_{m,n}$ CITE{Koelink}.   formula-special-values: $F_n(-1)=1$, $F_n(1)=(-1)^n$, $F_{2k+1}(0)=0$, and $F_{2k}(0)=\left(\binom{2k}{k}/4^k\right)^2$.   formula-touchard-q: Touchard's polynomials $Q_n$ satisfy $Q_n(x)=(-1)^n2^n n!\binom{2n}{n}^{-1}F_n(2x+1)$

   number-header: $F_n(x)$ Numbers:-- params:-    n: '0'-  number: '1'-- params:-    n: '1'-  number: -x-- params:-    n: '2'-  number: 3/4*x^2 + 1/4-- params:-    n: '3'-  number: -5/12*x^3 - 7/12*x-- params:-    n: '4'-  number: 35/192*x^4 + 65/96*x^2 + 9/64-- params:-    n: '5'-  number: -21/320*x^5 - 49/96*x^3 - 407/960*x-- params:-    n: '6'-  number: 77/3840*x^6 + 217/768*x^4 + 2303/3840*x^2 + 25/256-- params:-    n: '7'-  number: -143/26880*x^7 - 473/3840*x^5 - 2051/3840*x^3 - 3023/8960*x-- params:-    n: '8'-  number: 143/114688*x^8 + 2717/61440*x^6 + 8393/24576*x^4 + 231491/430080*x^2 + 1225/16384-- params:-    n: '9'-  number: -2431/9289728*x^9 - 10439/774144*x^7 - 186901/1105920*x^5 - 1242461/2322432*x^3+  '0': '1'+  '1': -x+  '2': 3/4*x^2 + 1/4+  '3': -5/12*x^3 - 7/12*x+  '4': 35/192*x^4 + 65/96*x^2 + 9/64+  '5': -21/320*x^5 - 49/96*x^3 - 407/960*x+  '6': 77/3840*x^6 + 217/768*x^4 + 2303/3840*x^2 + 25/256+  '7': -143/26880*x^7 - 473/3840*x^5 - 2051/3840*x^3 - 3023/8960*x+  '8': 143/114688*x^8 + 2717/61440*x^6 + 8393/24576*x^4 + 231491/430080*x^2 + 1225/16384+  '9': -2431/9289728*x^9 - 10439/774144*x^7 - 186901/1105920*x^5 - 1242461/2322432*x^3     - 1456787/5160960*x-- params:-    n: '10'-  number: 46189/928972800*x^10 + 31603/8847360*x^8 + 1502501/22118400*x^6 + 35140391/92897280*x^4+  '10': 46189/928972800*x^10 + 31603/8847360*x^8 + 1502501/22118400*x^6 + 35140391/92897280*x^4     + 7219663/14745600*x^2 + 3969/65536-- params:-    n: '11'-  number: -4199/486604800*x^11 - 155363/185794560*x^9 - 3550807/154828800*x^7 - 100919/491520*x^5+  '11': -4199/486604800*x^11 - 155363/185794560*x^9 - 3550807/154828800*x^7 - 100919/491520*x^5     - 489669869/928972800*x^3 - 11072177/45416448*x-- params:-    n: '12'-  number: 96577/70071091200*x^12 + 558467/3185049600*x^10 + 99058609/14863564800*x^8+  '12': 96577/70071091200*x^12 + 558467/3185049600*x^10 + 99058609/14863564800*x^8     + 143126893/1592524800*x^6 + 2561060957/6370099200*x^4 + 12272016991/27249868800*x^2     + 53361/1048576-- params:-    n: '13'-  number: -7429/36436967424*x^13 - 1166353/35035545600*x^11 - 721259/424673280*x^9-    - 366047179/11147673600*x^7 - 298657513/1274019840*x^5 - 2007748057/3892838400*x^3+  '13': -7429/36436967424*x^13 - 1166353/35035545600*x^11 - 721259/424673280*x^9 -+    366047179/11147673600*x^7 - 298657513/1274019840*x^5 - 2007748057/3892838400*x^3     - 225938249/1049624576*x-- params:-    n: '14'-  number: 7429/264505393152*x^14 + 453169/78479622144*x^12 + 9813709/25480396800*x^10+  '14': 7429/264505393152*x^14 + 453169/78479622144*x^12 + 9813709/25480396800*x^10     + 854817239/83235962880*x^8 + 19581791989/178362777600*x^6 + 23415419897/56056872960*x^4     + 8291566749497/19837904486400*x^2 + 184041/4194304-- params:-    n: '15'-  number: -215441/59513713459200*x^15 - 1567519/1700391813120*x^13 - 7347281/93428121600*x^11+  '15': -215441/59513713459200*x^15 - 1567519/1700391813120*x^13 - 7347281/93428121600*x^11     - 77774201/27745320960*x^9 - 2541919883/59454259200*x^7 - 964346539/3737124864*x^5     - 29944350920479/59513713459200*x^3 - 5213660383/26990346240*x-- params:-    n: '16'-  number: 6678671/15235510645555200*x^16 + 51921281/380887766138880*x^14 + 55546633/3805072588800*x^12+  '16': 6678671/15235510645555200*x^16 + 51921281/380887766138880*x^14 + 55546633/3805072588800*x^12     + 601919867/887850270720*x^10 + 756633004243/53271016243200*x^8 + 106932791825/837115969536*x^6     + 1630408326607681/3808877661388800*x^4 + 1503330129139/3847351173120*x^2 + 41409225/1073741824-- params:-    n: '17'-  number: -392863/7848596393164800*x^17 - 392863/20927899238400*x^15 - 193047809/77732197171200*x^13+  '17': -392863/7848596393164800*x^17 - 392863/20927899238400*x^15 - 193047809/77732197171200*x^13     - 21545712967/146495294668800*x^11 - 10412994631/2536715059200*x^9 - 14257719757/271790899200*x^7     - 1056803269448257/3808877661388800*x^5 - 1465895066081/2989699891200*x^3 - 20616229578539/117461986836480*x-- params:-    n: '18'-  number: 392863/72655578039582720*x^18 + 120608941/49861671203635200*x^16 + 2138353309/5484783832399872*x^14+  '18': 392863/72655578039582720*x^18 + 120608941/49861671203635200*x^16 + 2138353309/5484783832399872*x^14     + 130679958659/4520426235494400*x^12 + 402018701057/383551316951040*x^10 + 35163877306583/1917756584755200*x^8     + 153765704797379/1068464382935040*x^6 + 19872306199487837/45706531936665600*x^4     + 25381947965913563/69067648259850240*x^2 + 147744025/4294967296-- params:-    n: '19'-  number: -765049/1380455982752071680*x^19 - 1013173/3459789430456320*x^17 - 708125219/12465417800908800*x^15+  '19': -765049/1380455982752071680*x^19 - 1013173/3459789430456320*x^17 - 708125219/12465417800908800*x^15     - 12204206387/2350621642457088*x^13 - 103077892523/430516784332800*x^11 - 2133885882127/383551316951040*x^9     - 330325681439443/5342321914675200*x^7 - 4979084527103/16959752110080*x^5 - 164977336297110859/345338241299251200*x^3     - 159746458862789/991901222174720*x-- params:-    n: '20'-  number: 765049/14158522900021248000*x^20 + 97161223/2906223121583308800*x^18 + 628601477/81406810128384000*x^16+  '20': 765049/14158522900021248000*x^20 + 97161223/2906223121583308800*x^18 + 628601477/81406810128384000*x^16     + 9507718817/11080371378585600*x^14 + 35754344825627/723268197679104000*x^12 +     466021021397/313103115878400*x^10 + 3463743849565733/153420526780416000*x^8 +     6774303011494589/42738575317401600*x^6 + 202498114146683983/461305257394176000*x^4     + 3646744382722889053/10498282535497236480*x^2 + 2133423721/68719476736-- params:-    n: '21'-  number: -31367009/6243908598909370368000*x^21 - 322085629/89198694270133862400*x^19+  '21': -31367009/6243908598909370368000*x^21 - 322085629/89198694270133862400*x^19     - 28582995689/29062231215833088000*x^17 - 63454384003/483274659358310400*x^15     - 167029419962161/17950201633308672000*x^13 - 25715463653507/72326819767910400*x^11

     - 2974993444518794443/9687410405277696000*x^5 - 717829147207627871/1541705826891202560*x^3     - 1418473716888367/9522251732877312*x-- params:-    n: '22'-  number: 1348781387/3022051761872135258112000*x^22 + 14522925167/39247425478858899456000*x^20+  '22': 1348781387/3022051761872135258112000*x^22 + 14522925167/39247425478858899456000*x^20     + 11591257399/98364474884358144000*x^18 + 206781575749649/11057324206118141952000*x^16     + 1270433594146733/789808871865581568000*x^14 + 2438665039919047/31823800697880576000*x^12

     + 5629958371631519423/32788158294786048000*x^6 + 19459665047169300729497/44092786649088393216000*x^4     + 690769146779707001/2094895381233008640*x^2 + 7775536041/274877906944-- params:-    n: '23'-  number: -58642669/1544604233845758020812800*x^23 - 9910611061/274731978352012296192000*x^21+  '23': -58642669/1544604233845758020812800*x^23 - 9910611061/274731978352012296192000*x^21     - 104617157713/7849485095771779891200*x^19 - 12004620059563/4819859269333549056000*x^17     - 56826538238599/221146484122362839040*x^15 - 130598570544539659/8687897590521397248000*x^13

     - 32803315327103360761/413130794514304204800*x^7 - 9723003921965570488289/30525775372445810688000*x^5     - 8808931929092118964849/19400826125598893015040*x^3 - 11046045800444959/79640650856792064*x-- params:-    n: '24'-  number: 2756205443/889692038695156619988172800*x^24 + 4632770851/1381509376855833260851200*x^22+  '24': 2756205443/889692038695156619988172800*x^24 + 4632770851/1381509376855833260851200*x^22     + 23281139593/16310618380824477696000*x^20 + 14411235430649/46270648985602070937600*x^18     + 550310955331/14427791579676672000*x^16 + 2147743793640191/794322065419099176960*x^14

     + 226348509022273627708133/511670139576234541056000*x^4 + 2179670630301086872583/6938293502643724615680*x^2     + 457028729521/17592186044416-- params:-    n: '25'-  number: -2756205443/11348112738458630356992000000*x^25 - 1656479471243/5560575241844728874926080000*x^23+  '25': -2756205443/11348112738458630356992000000*x^25 - 1656479471243/5560575241844728874926080000*x^23     - 31902022434683/219785582681609836953600000*x^21 - 115233841510373/3139794038308711956480000*x^19     - 122287687837423567/23135324492801035468800000*x^17 - 167962769061635551/374247896207075573760000*x^15

     - 146951407131352194036882679/447711372129205223424000000*x^5 - 2027808057432009614301149/4576605137320764506112000*x^3     - 5515863820512167497/42475013790289100800*x-- params:-    n: '26'-  number: 2756205443/150418121788196747476992000000*x^26 + 85442368733/3372010642284850163220480000*x^24+  '26': 2756205443/150418121788196747476992000000*x^26 + 85442368733/3372010642284850163220480000*x^24     + 1766471361856801/125717353293880826737459200000*x^22 + 1335067388355667/326538579984106043473920000*x^20     + 16860570096995941/24551772931135792742400000*x^18 + 10761724689372507341/155687124822143438684160000*x^16

     + 832091428315848028933379503/4276100452172817235968000000*x^6 + 1895218962852989893389179839/4283702408532235577720832000*x^4     + 2166532273918912439933473/7215825242749473600307200*x^2 + 1690195005625/70368744177664-- params:-    n: '27'-  number: -146078888479/109654810783595428910727168000000*x^27 - 101979601391/49327400262526058889216000000*x^25+  '27': -146078888479/109654810783595428910727168000000*x^27 - 101979601391/49327400262526058889216000000*x^25     - 26853709631149/20691883486747944183398400000*x^23 - 399774433113748789/925736874254940633248563200000*x^21     - 7402376242562111927/88165416595708631737958400000*x^19 - 72386088100601202623/7291876560547330444492800000*x^17

     - 4056134907761623095522547901/42535946603192760926208000000*x^7 - 99293613370348285346747403533/295050931199924389281792000000*x^5     - 39429953640909693416639196869/91126033054231193198788608000*x^3 - 43108777127399077963/352869345334709452800*x-- params:-    n: '28'-  number: 146078888479/1563079484624342113927456358400000*x^28 + 110581718578603/682296600431260446555635712000000*x^26+  '28': 146078888479/1563079484624342113927456358400000*x^28 + 110581718578603/682296600431260446555635712000000*x^26     + 9820359993409/85688741027473839441838080000*x^24 + 3532366321225512853/81464844934434775725873561600000*x^22     + 143912353897091749/14848912268750927450603520000*x^20 + 77797380771717551/58276735748630013542400000*x^18

     + 7084753958163071397362313186151/16038181817544690002986795008000*x^4 + 42839111941531244071081529/148873868166199665858969600*x^2     + 25145962430625/1125899906842624-- params:-    n: '29'-  number: -5037203051/795250965861507391296425164800000*x^29 - 1365082026821/111648534616024436709104025600000*x^27+  '29': -5037203051/795250965861507391296425164800000*x^29 - 1365082026821/111648534616024436709104025600000*x^27     - 1093073062067/112748343457202420318208000000*x^25 - 3387061996901797/816423282567320192459735040000*x^23     - 4483528843445815841/4231939996594014323422003200000*x^21 - 485805822338397557/2878870745982322668994560000*x^19

     - 2066888717680880307668215360367953/6014318181579258751120048128000000*x^5 -     484436979986968686007887454439/1145584415538906428784771072000*x^3 - 18894620352267602119043/163731376235305186099200*x-- params:-    n: '30'-  number: 297194980009/715725869275356652166782648320000000*x^30 + 337492604417/379695421366236950751608832000000*x^28+  '30': 297194980009/715725869275356652166782648320000000*x^30 + 337492604417/379695421366236950751608832000000*x^28     + 5857798688429257/7443235641068295780606935040000000*x^26 + 2005386037239631439/5290422871036234847139083059200000*x^24     + 9728974914140864737/88468609883549077711355904000000*x^22 + 340566984512149582123/16927759986376057293688012800000*x^20