Boole polynomials $r_n(x)$

edit · history · discussion · files · short url · generating function polynomial number theory

Polynomials

$n$

$r_n(x)$

x - 1/2

x^2 - 2*x + 1/2

x^3 - 9/2*x^2 + 5*x - 3/4

x^4 - 8*x^3 + 20*x^2 - 16*x + 3/2

x^5 - 25/2*x^4 + 55*x^3 - 100*x^2 + 64*x - 15/4

x^6 - 18*x^5 + 245/2*x^4 - 390*x^3 + 574*x^2 - 312*x + 45/4

x^7 - 49/2*x^6 + 238*x^5 - 4655/4*x^4 + 2989*x^3 - 3773*x^2 + 1812*x - 315/8

x^8 - 32*x^7 + 420*x^6 - 2912*x^5 + 11424*x^4 - 25088*x^3 + 28160*x^2 - 12288*x + 315/2

x^9 - 81/2*x^8 + 690*x^7 - 6426*x^6 + 35553*x^5 - 118692*x^4 + 231020*x^3 - 236304*x^2 + 95616*x - 2835/4

10:

x^10 - 50*x^9 + 2145/2*x^8 - 12900*x^7 + 95403*x^6 - 447090*x^5 + 1317140*x^4 - 2327800*x^3 + 2208096*x^2 - 840960*x + 14175/4

11:

x^11 - 121/2*x^10 + 1595*x^9 - 96195/4*x^8 + 228723*x^7 - 2853543/2*x^6 + 5875925*x^5 - 15653770*x^4 + 25556476*x^3 - 22773168*x^2 + 8254080*x - 155925/8

12:

x^12 - 72*x^11 + 2288*x^10 - 42240*x^9 + 1003431/2*x^8 - 4009896*x^7 + 21900164*x^6 - 81251280*x^5 + 199180696*x^4 - 304256832*x^3 + 257182848*x^2 - 89441280*x + 467775/4

13:

x^13 - 169/2*x^12 + 3185*x^11 - 70642*x^10 + 1024023*x^9 - 40751139/4*x^8 + 71054555*x^7 - 348421216*x^6 + 1185340156*x^5 - 2708689412*x^4 + 3909228960*x^3 - 3158131392*x^2 + 1060369920*x - 6081075/8

14:

x^14 - 98*x^13 + 8645/2*x^12 - 113386*x^11 + 1968967*x^10 - 23837814*x^9 + 825403865/4*x^8 - 1288325038*x^7 + 5775067298*x^6 - 18255084848*x^5 + 39274578980*x^4 - 53961319776*x^3 + 41909678784*x^2 - 13649610240*x + 42567525/8

15:

x^15 - 225/2*x^14 + 5740*x^13 - 702975/4*x^12 + 3600142*x^11 - 104159055/2*x^10 + 547195220*x^9 - 33832688175/8*x^8 + 24071760713*x^7 - 99976371495*x^6 + 296634742040*x^5 - 605548602750*x^4 + 796866696144*x^3 - 597788238240*x^2 + 189550368000*x - 638512875/16

16:

x^16 - 128*x^15 + 7480*x^14 - 264320*x^13 + 6305572*x^12 - 107359616*x^11 + 1344731960*x^10 - 12585189760*x^9 + 88463817728*x^8 - 465377296384*x^7 + 1809483079040*x^6 - 5079891281920*x^5 + 9901384525824*x^4 - 12540751183872*x^3 + 9121468907520*x^2 - 2824077312000*x + 638512875/2

17:

x^17 - 289/2*x^16 + 9588*x^15 - 387260*x^14 + 10640742*x^13 - 210549794*x^12 + 3097588156*x^11 - 34487381500*x^10 + 292927290513*x^9 - 1898843734216*x^8 + 9330230497224*x^7 - 34242173230720*x^6 + 91545085129744*x^5 - 171239517345408*x^4 + 209588629900032*x^3 - 148266768107520*x^2 + 44927447040000*x - 10854718875/4

18:

x^18 - 162*x^17 + 24225/2*x^16 - 554472*x^15 + 17381922*x^14 - 395417484*x^13 + 6748170910*x^12 - 88080987384*x^11 + 888311328333*x^10 - 6944450916546*x^9 + 41961438905880*x^8 - 194200540659216*x^7 + 677192208310864*x^6 - 1733205672011808*x^5 + 3124469631836160*x^4 - 3707900113724928*x^3 + 2557806503546880*x^2 - 760034451456000*x + 97692469875/4

19:

x^19 - 361/2*x^18 + 15105*x^17 - 3111459/4*x^16 + 27591306*x^15 - 714917541*x^14 + 14003403370*x^13 - 211668327377*x^12 + 2498342766621*x^11 - 46299422056173/2*x^10 + 168390024439845*x^9 - 956555351152908*x^8 + 4198030176812536*x^7 - 13984853571016232*x^6 + 34416166165033680*x^5 - 60003715510162944*x^4 + 69237300674209536*x^3 - 46676149842216960*x^2 + 13622700994560000*x - 1856156927625/8

20:

x^20 - 200*x^19 + 18620*x^18 - 1071600*x^17 + 85391187/2*x^16 - 1249854960*x^15 + 27842109040*x^14 - 482286545600*x^13 + 6581910031151*x^12 - 71264075418120*x^11 + 613419165783060*x^10 - 4187758999866000*x^9 + 22519190501472976*x^8 - 94241205130638080*x^7 + 301278272240281280*x^6 - 715546448995564800*x^5 + 1210153230842121216*x^4 - 1360982737083248640*x^3 + 898326645239808000*x^2 - 257872110354432000*x + 9280784638125/4

21:

x^21 - 441/2*x^20 + 22715*x^19 - 1452360*x^18 + 64579746*x^17 - 8482338207/4*x^16 + 53295248710*x^15 - 1048453329420*x^14 + 16374285724181*x^13 - 409390476453231/2*x^12 + 2055807802430655*x^11 - 16583201106233580*x^10 + 107002281597887896*x^9 - 547785143426856888*x^8 + 2196180457652073200*x^7 - 6763401376470560640*x^6 + 15551049537098482176*x^5 - 25577540168993261568*x^4 + 28094078493014814720*x^3 - 18185377811779584000*x^2 + 5140559166898176000*x - 194896477400625/8

22:

x^22 - 242*x^21 + 54901/2*x^20 - 1939630*x^19 + 95697756*x^18 - 3502544892*x^17 + 394464395677/4*x^16 - 2185966124540*x^15 + 38721598493501*x^14 - 553218142768522*x^13 + 12812942943846591/2*x^12 - 60214420913596950*x^11 + 458434321443604726*x^10 - 2812005794823350312*x^9 + 13770290887865002568*x^8 - 53097568431508252960*x^7 + 158034796840720970016*x^6 - 352726524428780380032*x^5 + 565446257760737346048*x^4 - 607666766286379345920*x^3 + 386283966709745664000*x^2 - 107637093007589376000*x + 2143861251406875/8

23:

x^23 - 529/2*x^22 + 32890*x^21 - 10223983/4*x^20 + 139202371*x^19 - 5646571392*x^18 + 176996624200*x^17 - 35121409928011/8*x^16 + 87521026854151*x^15 - 2833097687230349/2*x^14 + 18725054488924210*x^13 - 810724326981796773/4*x^12 + 1795696721692314661*x^11 - 12979395797898815417*x^10 + 76052296290062941420*x^9 - 357680323431513666892*x^8 + 1330930253403269562736*x^7 - 3839087540578973896752*x^6 + 8337077896288121513280*x^5 - 13051316816951548177152*x^4 + 13744314486064293166080*x^3 - 8590742396497529856000*x^2 + 2361827297364885504000*x - 49308808782358125/16

24:

x^24 - 288*x^23 + 39100*x^22 - 3327456*x^21 + 199095820*x^20 - 8905098048*x^19 + 309035300200*x^18 - 8524549826496*x^17 + 379909253385695/2*x^16 - 3456298360894368*x^15 + 51700392572086300*x^14 - 638057368168404576*x^13 + 6502557037952959840*x^12 - 54630072028881980928*x^11 + 376737204554219502400*x^10 - 2117376815642492929536*x^9 + 9596877596703609815680*x^8 - 34559939395891184529408*x^7 + 96848583021250536960000*x^6 - 205050245511374911143936*x^5 + 313991700089360088858624*x^4 - 324471624108838378536960*x^3 + 199627875410463424512000*x^2 - 54193944307263602688000*x + 147926426347074375/4

25:

x^25 - 625/2*x^24 + 46150*x^23 - 4283750*x^22 + 280403695*x^21 - 13765540250*x^20 + 526222239400*x^19 - 16054166232500*x^18 + 397443552698335*x^17 - 32300076622100375/4*x^16 + 135650640887353150*x^15 - 1892454976221293750*x^14 + 21966662302344164065*x^13 - 212052891711167871500*x^12 + 1697821428186239398900*x^11 - 11217591236595890630000*x^10 + 60685906154938310612080*x^9 - 265862875486457771984000*x^8 + 928909407504189451454400*x^7 - 2534321379344817569760000*x^6 + 5240631425320146771661824*x^5 - 7861457660255664228864000*x^4 + 7981390674638577451008000*x^3 - 4838135659029511372800000*x^2 + 1297872705574034472960000*x - 3698160658676859375/8

26:

x^26 - 338*x^25 + 108225/2*x^24 - 5458700*x^23 + 389374245*x^22 - 20892352910*x^21 + 875781599550*x^20 - 29404826937200*x^19 + 804371465389635*x^18 - 18139519503621230*x^17 + 1359747273778449075/4*x^16 - 5320831185010144700*x^15 + 69747861617871300615*x^14 - 766111168673207677970*x^13 + 7040906338489293804300*x^12 - 53953692941719624968200*x^11 + 342756786526857398432880*x^10 - 1790286084527039412067040*x^9 + 7600675184571066912828800*x^8 - 25822290515522389847587200*x^7 + 68716533577390235233242624*x^6 - 139002353848999580708800512*x^5 + 204538480185068310647808000*x^4 - 204238250318652668411904000*x^3 + 122085892379084851445760000*x^2 - 32383555215793434132480000*x + 48076088562799171875/8

27:

x^27 - 729/2*x^26 + 63063*x^25 - 27562275/4*x^24 + 533705445*x^23 - 62358960105/2*x^22 + 1427301067335*x^21 - 52504558402725*x^20 + 1579294851469335*x^19 - 78638236029696915/2*x^18 + 817136669003787405*x^17 - 114044165659754053425/8*x^16 + 209468862114969832815*x^15 - 5191870621371705263835/2*x^14 + 27120878050643487300645*x^13 - 238323937347250877485350*x^12 + 1754235328922087089658580*x^11 - 10748716534354297579268760*x^10 + 54348921861696023666801840*x^9 - 224111486094102343104825600*x^8 + 741779451272625971214469824*x^7 - 1928576595408203625811400448*x^6 + 3821599085879018504848499712*x^5 - 5522577449692134922781184000*x^4 + 5428890968370642427428864000*x^3 - 3202614106265207774085120000*x^2 + 840469456539816996372480000*x - 1298054391195577640625/16

28:

x^28 - 392*x^27 + 73080*x^26 - 8622432*x^25 + 1445604615/2*x^24 - 45815154840*x^23 + 2281686072900*x^22 - 91585587933840*x^21 + 3015794189292885*x^20 - 82493132723721720*x^19 + 1891143595179165600*x^18 - 36557122228552336320*x^17 + 2392828079202044924835/4*x^16 - 8303119558622155475880*x^15 + 97788629451961052482500*x^14 - 975980180872644191833680*x^13 + 8230731545066810537819880*x^12 - 58379027324790490075517760*x^11 + 345942588940067670625131200*x^10 - 1697248889621824044422076160*x^9 + 6811303354395115885843794624*x^8 - 22001636001383189170510353408*x^7 + 55969531231695132081030174720*x^6 - 108780512510273124259466477568*x^5 + 154543073998988964006770688000*x^4 - 149694142021195004168011776000*x^3 + 87210162045823696021094400000*x^2 - 22655441841975790109982720000*x + 9086380738369043484375/8

29:

x^29 - 841/2*x^28 + 84245*x^27 - 10702566*x^26 + 968067009*x^25 - 265445094915/4*x^24 + 3583104898425*x^23 - 156352674908820*x^22 + 5613615840447615*x^21 - 335944941908460885/2*x^20 + 4227550501953480975*x^19 - 90077717395793255310*x^18 + 1631989850359931268555*x^17 - 201649517705650852200135/8*x^16 + 332216322394732548897075*x^15 - 3734697998083125868419240*x^14 + 35741969913440429384103540*x^13 - 290202839945098374703385340*x^12 + 1988908969825245193578710000*x^11 - 11425427132044071063581572480*x^10 + 54502042988220363692928151744*x^9 - 213245413696470732224777412192*x^8 + 673261444055896440350288417280*x^7 - 1677981177254994157317300931584*x^6 + 3202331929725331602214547521536*x^5 - 4476919953141720090741537792000*x^4 + 4276249340857606632772288512000*x^3 - 2461896026741963985615912960000*x^2 + 633392251320362817096253440000*x - 263505041412702261046875/16

30:

x^30 - 450*x^29 + 193285/2*x^28 - 13184850*x^27 + 1283224509*x^26 - 94849856010*x^25 + 22138971308025/4*x^24 - 261659569772250*x^23 + 10205693517740565*x^22 - 332730812462998950*x^21 + 18308094685695994725/2*x^20 - 213979995041323533750*x^19 + 4270105261687986490755*x^18 - 72967471444829662066350*x^17 + 8554779074797408070994225/8*x^16 - 13442819281997306381286750*x^15 + 144796850521895004244070490*x^14 - 1333104242676061632139490400*x^13 + 10450315416499517743279397900*x^12 - 69372873274649477280225192000*x^11 + 387141869249344586348529566944*x^10 - 1798878248453460543689398041600*x^9 + 6872882864157406185270778789920*x^8 - 21238238574273227071355450150400*x^7 + 51920998414618311182340336116736*x^6 - 97396444777501121858977799946240*x^5 + 134104799603211103700116706304000*x^4 - 126405932954899712982641786880000*x^3 + 71956440380989265050731479040000*x^2 - 18342645763545144210987417600000*x + 3952575621190533915703125/16

Definition

For an integer $n\geq 0$, Jordan's Boole polynomial $r_n(x)$ is the polynomial in $\mathbb{Q}[x]$ defined by $\sum_{n=0}^{\infty}r_n(x)\frac{t^n}{n!} =\frac{2(1+t)^x}{2+t}$ [2].

Parameters

$n$

— index ($n\geq 0$)

Formulas

(1)

$\sum_{n=0}^{\infty}r_n(x)\frac{t^n}{n!}=\frac{2(1+t)^x}{2+t}$.

(2)

$r_n(x)=n!\sum_{j=0}^{n}\binom{x}{j}\left(-\frac12\right)^{n-j}$, where $\binom{x}{j}=x(x-1)\cdots(x-j+1)/j!$.

(3)

$r_n(x+1)+r_n(x)=2(x)_n$, where $(x)_n=x(x-1)\cdots(x-n+1)$ and $(x)_0=1$.

(4)

$r_n(x)=\sum_{m=0}^{n}s(n,m)E_m(x)$, where $s(n,m)$ is the signed Stirling number of the first kind and $E_m(x)$ is the Euler polynomial.

(5)

$r_n(0)=(-1)^n n!/2^n$.

Comments

(6)

The name Boole polynomials is also used for the two-variable family $s_n(x;\lambda)$, defined by $\sum_{n=0}^{\infty}s_n(x;\lambda)t^n/n! =(1+t)^x/(1+(1+t)^\lambda)$ [1] [2]. This table stores Jordan's one-variable normalisation, for which $r_n(x)=2s_n(x;1)$.

(7)

The Euler polynomials are the derivative analogue of this finite-difference family. Replacing $2/(2+t)$ by $t/\log(1+t)$ gives the Bernoulli polynomials of the second kind.

Programs

(P1)

Sage

import numberdb.sage as numberdb
from sage.arith.misc import factorial
from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing
from sage.rings.rational_field import QQ

R = PolynomialRing(QQ, 'x')
x = R.gen()

def binomial_x(j):
    value = R.one()
    for m in range(j):
        value *= x - QQ(m)
        value /= QQ(m + 1)
    return value

def boole_polynomial(n):
    coefficient = R.zero()
    for j in range(n + 1):
        coefficient += binomial_x(j) * QQ((-1)**(n - j)) / QQ(2**(n - j))
    return coefficient * QQ(factorial(n))

boole_polynomial(31)      # the next one after this table

Links

[1]

Wikipedia: Boole polynomials

[2]

MathWorld: Boole Polynomial

Similar tables

Euler polynomials — give the Stirling transform in (4)

Bernoulli polynomials of the second kind — are the Gregory-Newton companion with $t/\log(1+t)$ in place of $2/(2+t)$

Data properties

Entries are of type: rational polynomial

How well the digits are known: exact

Table is complete: no (it holds every index $n$ with $0\leq n\leq30$)

How they were obtained:

The generator computes (2) over Sage's rational polynomial ring, with every division made in Sage's rational field.

Before the draft was created, the entries were checked against (1), against the polynomials $r_0$ through $r_3$ printed by [2], against the printed values of $s_0(x;1)$ through $s_2(x;1)$ with $r_n(x)=2s_n(x;1)$, against (3) and (5) on every entry, and against (4) with the Euler polynomials computed from their derivative and reflection identities.