Stirling convolution polynomials $\sigma_n(x)$

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Polynomials

$n$

$\sigma_n(x)$

1/2

1/8*x - 1/24

1/48*x^2 - 1/48*x

1/384*x^3 - 1/192*x^2 + 1/1152*x + 1/2880

1/3840*x^4 - 1/1152*x^3 + 1/2304*x^2 + 1/5760*x

1/46080*x^5 - 1/9216*x^4 + 1/9216*x^3 + 13/414720*x^2 - 1/69120*x - 1/181440

1/645120*x^6 - 1/92160*x^5 + 1/55296*x^4 + 1/829440*x^3 - 1/138240*x^2 - 1/362880*x

1/10321920*x^7 - 1/1105920*x^6 + 1/442368*x^5 - 1/1658880*x^4 - 67/39813120*x^3 - 1/2580480*x^2 + 101/348364800*x + 1/9676800

1/185794560*x^8 - 1/15482880*x^7 + 1/4423680*x^6 - 1/6220800*x^5 - 19/79626240*x^4 + 1/27869184*x^3 + 101/696729600*x^2 + 1/19353600*x

10:

1/3715891200*x^9 - 1/247726080*x^8 + 1/53084160*x^7 - 19/796262400*x^6 - 7/318504960*x^5 + 149/6688604160*x^4 + 67/2090188800*x^3 + 47/8360755200*x^2 - 13/2090188800*x - 1/479001600

11:

1/81749606400*x^10 - 1/4459069440*x^9 + 1/743178240*x^8 - 29/11147673600*x^7 - 11/9555148800*x^6 + 289/66886041600*x^5 + 11/2786918400*x^4 - 1/668860416*x^3 - 13/4180377600*x^2 - 1/958003200*x

12:

1/1961990553600*x^11 - 1/89181388800*x^10 + 1/11890851840*x^9 - 61/267544166400*x^8 + 1/76441190400*x^7 + 143/267544166400*x^6 + 1151/4815794995200*x^5 - 481/802632499200*x^4 - 793/1203948748800*x^3 - 311/3678732288000*x^2 + 7999/57940033536000*x + 691/15692092416000

13:

1/51011754393600*x^12 - 1/1961990553600*x^11 + 1/214035333120*x^10 - 1/59454259200*x^9 + 11/1070176665600*x^8 + 541/11236854988800*x^7 - 389/48157949952000*x^6 - 829/8026324992000*x^5 - 169/2407897497600*x^4 + 47/1051066368000*x^3 + 7999/115880067072000*x^2 + 691/31384184832000*x

14:

1/1428329123020800*x^13 - 1/47087773286400*x^12 + 1/4280706662400*x^11 - 23/21403533312000*x^10 + 17/12842119987200*x^9 + 59/17978967982080*x^8 - 713/192631799808000*x^7 - 937/82556485632000*x^6 - 11/6421059993600*x^5 + 24151/1589212348416000*x^4 + 8689/618027024384000*x^3 + 22271/18077290463232000*x^2 - 2357/753220435968000*x - 1/1046139494400

15:

1/42849873690624000*x^14 - 1/1224282105446400*x^13 + 1/94175546572800*x^12 - 257/4237899595776000*x^11 + 1/8561413324800*x^10 + 89/539369039462400*x^9 - 3959/8090535591936000*x^8 - 6901/8090535591936000*x^7 + 389/577895399424000*x^6 + 7699/3178424696832000*x^5 + 14209/11124486438912000*x^4 - 1259/1032988026470400*x^3 - 2357/1506440871936000*x^2 - 1/2092278988800*x

16:

1/1371195958099968000*x^15 - 1/34279898952499200*x^14 + 1/2260213117747200*x^13 - 13/4237899595776000*x^12 + 17/2054739197952000*x^11 + 1/199766310912000*x^10 - 2837/64724284735488000*x^9 - 13/323621423677440*x^8 + 2047/15850845241344000*x^7 + 1043/4358982441369600*x^6 - 28691/1067950698135552000*x^5 - 130943/347083976894054400*x^4 - 10662539/34708397689405440000*x^3 - 5281/333734593167360000*x^2 + 52037/723091618529280000*x + 3617/170729965486080000

17:

1/46620662575398912000*x^16 - 1/1028396968574976000*x^15 + 1/58765541061427200*x^14 - 31/220370778980352000*x^13 + 29/58119765884928000*x^12 - 1/14832648585216000*x^11 - 131/43149523156992000*x^10 - 29/97086427103232000*x^9 + 21487/1553382833651712000*x^8 + 153859/10679506981355520000*x^7 - 256303/10679506981355520000*x^6 - 12731/225379205775360000*x^5 - 32891/1416669293445120000*x^4 + 25177/788827220213760000*x^3 + 52037/1446183237058560000*x^2 + 3617/341459930972160000*x

18:

1/1678343852714360832000*x^17 - 1/32908702994399232000*x^16 + 1/1645435149719961600*x^15 - 437/74044581737398272000*x^14 + 43/1627353444777984000*x^13 - 263/11391474113445888000*x^12 - 1319/7766914168258560000*x^11 + 1201/7766914168258560000*x^10 + 2189/2071177111535616000*x^9 + 2265503/9227094031891169280000*x^8 - 235759/64077041888133120000*x^7 - 25040803/4998009267274383360000*x^6 + 530941/277667181515243520000*x^5 + 473129/51000094564024320000*x^4 + 9473/1388335907576217600*x^3 + 7132679/55759040888029839360000*x^2 - 1295681/774431123444858880000*x - 43867/91963695909076992000

19:

1/63777066403145711616000*x^18 - 1/1118895901809573888000*x^17 + 1/49363054491598848000*x^16 - 13/56957370567229440000*x^15 + 53/42311189564227584000*x^14 - 73/35541399233951170560*x^13 - 1319/170872111701688320000*x^12 + 3131/170872111701688320000*x^11 + 2263/37281188007641088000*x^10 - 1019281/18454188063782338560000*x^9 - 6511/18689137217372160000*x^8 - 1391087/6361102703803760640000*x^7 + 170827/238000441298780160000*x^6 + 1314421/999601853454876672000*x^5 + 869/2103539253903360000*x^4 - 7023661/8578313982773821440000*x^3 - 1295681/1548862246889717760000*x^2 - 43867/183927391818153984000*x

20:

1/2551082656125828464640000*x^19 - 1/40280252465144659968000*x^18 + 1/1579617743731163136000*x^17 - 97/11847133077983723520000*x^16 + 383/7108279846790234112000*x^15 - 3281/24878979463765819392000*x^14 - 751/2733953787227013120000*x^13 + 5639/4100930680840519680000*x^12 + 3779/1491247520305643520000*x^11 - 82483/10545250322161336320000*x^10 - 71800261/3100303594715432878080000*x^9 + 417371/49757958927531638784000*x^8 + 3335117/34272063547024343040000*x^7 + 1384769/13328024712731688960000*x^6 - 1216651/17136031773512171520000*x^5 - 40520646403/178428930841695485952000000*x^4 - 513310799/3345542453281790361600000*x^3 + 24989087/14125623691634225971200000*x^2 + 152388293/3884546515199412142080000*x + 174611/16057153253965824000000

21:

1/107145471557284795514880000*x^20 - 1/1530649593675497078784000*x^19 + 1/53707003286859546624000*x^18 - 331/1208407573954339799040000*x^17 + 151/71082798467902341120000*x^16 - 1747/248789794637658193920000*x^15 - 4099/639745186211121070080000*x^14 + 7/87040161389268172800*x^13 + 5767/98422336340172472320000*x^12 - 3118319/4871905648838537379840000*x^11 - 6516997/6200607189430865756160000*x^10 + 6636713/2686929782086708494336000*x^9 + 140007961/16793311138041928089600000*x^8 + 21034703/8396655569020964044800000*x^7 - 23931041/1199522224145852006400000*x^6 - 32765421289/1070573585050172915712000000*x^5 - 46865639/6691084906563580723200000*x^4 + 586486687/28251247383268451942400000*x^3 + 152388293/7769093030398824284160000*x^2 + 174611/32114306507931648000000*x

22:

1/4714400748520531002654720000*x^21 - 1/61225983747019883151360000*x^20 + 1/1933452118326943678464000*x^19 - 83/9667260591634718392320000*x^18 + 1/12924145175982243840000*x^17 - 1949/5970955071303796654080000*x^16 + 41/1628442292173762723840000*x^15 + 23273/5970955071303796654080000*x^14 - 599/393689345360689889280000*x^13 - 7601791/194876225953541495193600000*x^12 - 2698721/124012143788617315123200000*x^11 + 174715687/690924801108010755686400000*x^10 + 1151822417/2418236803878037644902400000*x^9 - 99961079/201519733656503137075200000*x^8 - 17846993/7197133344875112038400000*x^7 - 2022508057/951620964489042591744000000*x^6 + 288529211/129766495157596717056000000*x^5 + 33855195523/6102269434785985619558400000*x^4 + 116927697319/33562481891322920907571200000*x^3 - 23163719/166480564937117663232000000*x^2 - 269839961/291340988639955910656000000*x - 77683/310224200866619719680000

23:

1/216862434431944426122117120000*x^22 - 1/2571491317374835092357120000*x^21 + 1/73471180496423859781632000*x^20 - 31/122451967494039766302720000*x^19 + 19/7250445443726038794240000*x^18 - 251/18455679311302644203520000*x^17 + 2437/179128652139113899622400000*x^16 + 85961/537385956417341698867200000*x^15 - 31/112482670103054254080000*x^14 - 1360769/723825982113154125004800000*x^13 + 10904357/8184801490048742798131200000*x^12 + 1869962999/106402419370633656375705600000*x^11 + 70346681/4836473607756075289804800000*x^10 - 32667203/403039467313006274150400000*x^9 - 81217/419832778451048202240000*x^8 + 5533739/17129177360802766651392000000*x^7 + 13708906129/25693766041204149977088000000*x^6 + 8698892071/12204538869571971239116800000*x^5 + 1029732337/9589280540377977402163200000*x^4 - 132947/254362971638071296000000*x^3 - 269839961/582681977279911821312000000*x^2 - 77683/620448401733239439360000*x

24:

1/10409396852733332453861621760000*x^23 - 1/113145617964492744063713280000*x^22 + 1/2938847219856954391265280000*x^21 - 233/33062031223390736901734400000*x^20 + 1/12052688529830298255360000*x^19 - 503/974459867636779613945856000*x^18 + 2117/2149543825669366795468800000*x^17 + 2983/537385956417341698867200000*x^16 - 5233/264559240082383605596160000*x^15 - 30454883/425609677482534625502822400000*x^14 + 12270233/65478411920389942385049600000*x^13 + 27827329/30400691248752473250201600000*x^12 - 275267/723586700796753082318848000*x^11 - 164796869/23215073317229161391063040000*x^10 - 3192584459/348226099758437420865945600000*x^9 + 14913396679/822200513318532799266816000000*x^8 + 152506913641/2466601539955598397800448000000*x^7 + 166493270833/3905452438263030796517376000000*x^6 - 124647957701681/1933198956940200244276101120000000*x^5 - 4363941670597/32219982615670004071268352000000*x^4 - 2697566659399/33830981746453504274831769600000*x^3 + 576654122683/108071191690059805322379264000000*x^2 + 19311547193251/878078432481735918244331520000000*x + 236364091/40651779281561848066867200000

25:

1/520469842636666622693081088000000*x^24 - 1/5204698426366666226930810880000*x^23 + 1/123431583233992084433141760000*x^22 - 43/231434218563735158312140800000*x^21 + 29/11755388879427817565061120000*x^20 - 257/14242105757768317434593280000*x^19 + 3727/73084490072758471045939200000*x^18 + 443/2810941925875325809459200000*x^17 - 757/721525200224682560716800000*x^16 - 1687193/851219354965069251005644800000*x^15 + 22522321/1702438709930138502011289600000*x^14 + 58725649/1659877742181885039461007360000*x^13 - 2707316981/30643896778742493036203212800000*x^12 - 22663379/52115470712147097000345600000*x^11 - 59924939/696452199516874841731891200000*x^10 + 136265093773/57554035932297295948677120000000*x^9 + 151497696367/34532421559378377569206272000000*x^8 - 223281317659/164029002407047293453729792000000*x^7 - 53916176103281/3866397913880400488552202240000000*x^6 - 411768291581/24784602012053849285591040000000*x^5 - 25405683763/19900577497913826044018688000000*x^4 + 149043299257/11375914914743137402355712000000*x^3 + 19311547193251/1756156864963471836488663040000000*x^2 + 236364091/81303558563123696133734400000*x

Definition

For an integer $n\geq 1$, the Stirling convolution polynomial $\sigma_n(x)$ is the polynomial in $\mathbb{Q}[x]$ determined by $[z^n]\left(\frac{ze^z}{e^z-1}\right)^x=x\sigma_n(x)$ [3] [1].

Parameters

$n$

— index ($n\geq 1$)

Formulas

(1)

$\left(\frac{ze^z}{e^z-1}\right)^x =\sum_{n=0}^{\infty}x\sigma_n(x)z^n$, with $\sigma_0(x)=1/x$.

(2)

$S_n(x)=n!(x+1)\sigma_n(x+1)$, where $S_n(x)$ is the Stirling polynomial in the Sheffer-sequence convention.

(3)

$(x+1)\sigma_n(x+1)=(x-n)\sigma_n(x)+x\sigma_{n-1}(x)$ for $n\geq1$, with $\sigma_0(x)=1/x$ [3].

(4)

If $m$ and $n$ are integers with $m>n\geq1$, then $m(m-1)\cdots(m-n)\sigma_n(m)=\left[{m\atop m-n}\right]$, where $\left[{a\atop b}\right]$ is an unsigned Stirling number of the first kind [3] [6].

(5)

If $m$ and $n$ are integers with $m>n\geq1$, then $m(m-1)\cdots(m-n)(-1)^{n+1}\sigma_n(n-m)=\left\{{m\atop m-n}\right\}$, where $\left\{{a\atop b}\right\}$ is a Stirling number of the second kind [7]. These numbers are the coefficients of the Touchard polynomials.

(6)

$\sigma_n(0)=-B_n/(n\,n!)$, where $B_n$ is the Bernoulli number with $B_1=-\tfrac12$.

(7)

If $B_n^{(a)}$ is defined by $\left(\frac{t}{e^t-1}\right)^a=\sum_{n=0}^{\infty}B_n^{(a)}t^n/n!$, then $B_n^{(x)}=(-1)^n n!x\sigma_n(x)$.

Comments

(8)

The same generating function gives $\sigma_0(x)=1/x$, which is not a polynomial. This table starts at $n=1$ and stores the polynomial entries.

(9)

Knuth [1] calls these convolution polynomials, and Concrete Mathematics [2] calls them Stirling polynomials. The same name is also used for the Sheffer sequence $S_n(x)$ of Stirling polynomials, related by (2), and for the Gessel-Stanley polynomials formed from $\left\{{n+k\atop n}\right\}$ and $\left[{n\atop n-k}\right]$ as polynomials in $n$ for fixed $k$.

Programs

(P1)

Sage

R.<x> = QQ[]

def stirling_convolution_polynomial(n):
    S.<z> = PowerSeriesRing(R, default_prec=n + 2)
    f = (x * (z / (1 - exp(-z))).log()).exp()
    return R(f[n] / x)

stirling_convolution_polynomial(26)      # the next one after this table

References

[1]

Donald E. Knuth, Convolution polynomials, The Mathematica Journal 2 (1992), no. 4, 67-78. (arXiv)

[2]

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, second edition, Addison-Wesley, 1994, Sections 6.2 and 7.4.

Links

[3]

Wikipedia: Stirling polynomials

[4]

OEIS A100655: coefficients of Bernoulli polynomials after multiplication by common denominators

[5]

OEIS A001898: denominators of Bernoulli polynomials

[6]

OEIS A132393: triangle of unsigned Stirling numbers of the first kind

[7]

OEIS A008277: triangle of Stirling numbers of the second kind

Similar tables

Stirling polynomials — hold the same polynomials in the Sheffer-sequence convention, related by (2)

Bernoulli numbers — give the value at $x=0$ in (6)

Touchard polynomials — have as coefficients the Stirling numbers of the second kind in (5)

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 $1\leq n\leq25$)

How they were obtained:

The generator expands $\left(z/(1-e^{-z})\right)^x$ as $(1+b(z))^x$ over Sage's rational polynomial ring and divides the coefficient of $z^n$ by $x$. Every division is made in Sage's rational field.

Before the draft was created, the entries were checked against the polynomials $\sigma_1$ through $\sigma_{10}$ printed by [3], against (3), (4), (5), (6), and (2) on every entry, using the stored values of Stirling polynomials for the last comparison. The Noerlund relation in (7) was checked against OEIS A100655 [4] and A001898 [5] through $n=8$.