back to table · edit · history · where entries came from · files
which is not a polynomial. This table starts at $n=1$ and stores the polynomial entries.- comment-convention: Knuth writes these as Stirling polynomials CITE{Concrete}. This- title includes "convolution" to distinguish them from HREF{Stirling_polynomials}[Stirling- polynomials] $S_k(x)$ and from the Gessel-Stanley polynomials $S(n+k,n)$ and $c(n,n-k)$.+ comment-convention: Knuth CITE{Knuth} calls these convolution polynomials, and Concrete+ Mathematics CITE{Concrete} calls them Stirling polynomials. The same name is also+ used for the Sheffer sequence $S_n(x)$ of HREF{Stirling_polynomials}[Stirling+ polynomials], related by CITE{formula-stirling-polynomials}, 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 $k$. Formulas: formula-generating-function: $\left(\frac{ze^z}{e^z-1}\right)^x =\sum_{n=0}^{\infty}x\sigma_n(x)z^n$,
formula-first-kind: 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 CITE{Wiki}.+ the first kind CITE{Wiki} CITE{OEISFirstKind}. formula-second-kind: 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. These numbers are the coefficients of the HREF{Touchard_polynomials}[Touchard+ kind CITE{OEISSecondKind}. These numbers are the coefficients of the HREF{Touchard_polynomials}[Touchard polynomials]. formula-bernoulli-value: $\sigma_n(0)=-B_n/(n\,n!)$, where $B_n$ is the HREF{Bernoulli_numbers}[Bernoulli
program-sage: language: Sage- code: "import numberdb.sage as numberdb\nfrom sage.arith.misc import factorial\n\- from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing\n\- from sage.rings.rational_field import QQ\n\nR = PolynomialRing(QQ, 'x')\nx =\- \ R.gen()\n\ndef mul(a, b, order):\n out = [R.zero() for _ in range(order\- \ + 1)]\n for i, ai in enumerate(a):\n for j, bj in enumerate(b[:order\- \ + 1 - i]):\n out[i + j] += ai * bj\n return out\n\ndef base_series(order):\n\- \ a = [QQ(0) for _ in range(order + 1)]\n a[0] = QQ(1)\n for r in range(2,\- \ order + 2):\n total = QQ(0)\n for m in range(2, r + 1):\n \- \ total += a[r - m] * QQ((-1)**(m + 1)) / QQ(factorial(m))\n \- \ a[r - 1] = -total\n return a\n\ndef binomial_x(m):\n value = R.one()\n\- \ for j in range(m):\n value *= x - QQ(j)\n value /= QQ(j +\- \ 1)\n return value\n\ndef stirling_convolution_polynomial(n):\n base\- \ = [R(c) for c in base_series(n)]\n base[0] = R.zero()\n coefficients\- \ = [R.zero() for _ in range(n + 1)]\n coefficients[0] = R.one()\n power\- \ = [R.zero() for _ in range(n + 1)]\n power[0] = R.one()\n for m in range(1,\- \ n + 1):\n power = mul(power, base, n)\n factor = binomial_x(m)\n\- \ for j in range(n + 1):\n coefficients[j] += factor*power[j]\n\- \ quotient, remainder = coefficients[n].quo_rem(x)\n if remainder:\n \- \ raise ArithmeticError('coefficient is not divisible by x')\n return\- \ quotient\n\nstirling_convolution_polynomial(26) # the next one after\+ code: "R.<x> = QQ[]\n\ndef stirling_convolution_polynomial(n):\n S.<z> = PowerSeriesRing(R,\+ \ default_prec=n + 2)\n f = (x * (z / (1 - exp(-z))).log()).exp()\n return\+ \ R(f[n] / x)\n\nstirling_convolution_polynomial(26) # the next one after\ \ this table" Similar tables: - table: HREF{Stirling_polynomials}[Stirling polynomials]- relation: are the same Stirling family in the Sheffer-sequence normalisation of- CITE{formula-stirling-polynomials}+ relation: hold the same polynomials in the Sheffer-sequence convention, related+ by CITE{formula-stirling-polynomials} - table: HREF{Bernoulli_numbers}[Bernoulli numbers] relation: give the value at $x=0$ in CITE{formula-bernoulli-value} - table: HREF{Touchard_polynomials}[Touchard polynomials]- relation: have Stirling numbers of the second kind as coefficients, which give CITE{formula-second-kind}+ relation: have as coefficients the Stirling numbers of the second kind in CITE{formula-second-kind} Links: Wiki:
url: https://oeis.org/A001898 OEISFirstKind:- title: 'OEIS A008275: triangle of unsigned Stirling numbers of the first kind'- url: https://oeis.org/A008275+ title: 'OEIS A132393: triangle of unsigned Stirling numbers of the first kind'+ url: https://oeis.org/A132393 OEISSecondKind: title: 'OEIS A008277: triangle of Stirling numbers of the second kind'
number-header: $\sigma_n(x)$ Numbers:-- params:- n: '1'- number: 1/2-- params:- n: '2'- number: 1/8*x - 1/24-- params:- n: '3'- number: 1/48*x^2 - 1/48*x-- params:- n: '4'- number: 1/384*x^3 - 1/192*x^2 + 1/1152*x + 1/2880-- params:- n: '5'- number: 1/3840*x^4 - 1/1152*x^3 + 1/2304*x^2 + 1/5760*x-- params:- n: '6'- number: 1/46080*x^5 - 1/9216*x^4 + 1/9216*x^3 + 13/414720*x^2 - 1/69120*x - 1/181440-- params:- n: '7'- number: 1/645120*x^6 - 1/92160*x^5 + 1/55296*x^4 + 1/829440*x^3 - 1/138240*x^2 -- 1/362880*x-- params:- n: '8'- number: 1/10321920*x^7 - 1/1105920*x^6 + 1/442368*x^5 - 1/1658880*x^4 - 67/39813120*x^3+ '1': 1/2+ '2': 1/8*x - 1/24+ '3': 1/48*x^2 - 1/48*x+ '4': 1/384*x^3 - 1/192*x^2 + 1/1152*x + 1/2880+ '5': 1/3840*x^4 - 1/1152*x^3 + 1/2304*x^2 + 1/5760*x+ '6': 1/46080*x^5 - 1/9216*x^4 + 1/9216*x^3 + 13/414720*x^2 - 1/69120*x - 1/181440+ '7': 1/645120*x^6 - 1/92160*x^5 + 1/55296*x^4 + 1/829440*x^3 - 1/138240*x^2 - 1/362880*x+ '8': 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-- params:- n: '9'- number: 1/185794560*x^8 - 1/15482880*x^7 + 1/4423680*x^6 - 1/6220800*x^5 - 19/79626240*x^4+ '9': 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-- params:- n: '10'- number: 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-- params:- n: '11'- number: 1/81749606400*x^10 - 1/4459069440*x^9 + 1/743178240*x^8 - 29/11147673600*x^7+ '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-- params:- n: '12'- number: 1/1961990553600*x^11 - 1/89181388800*x^10 + 1/11890851840*x^9 - 61/267544166400*x^8+ '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-- params:- n: '13'- number: 1/51011754393600*x^12 - 1/1961990553600*x^11 + 1/214035333120*x^10 - 1/59454259200*x^9+ '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-- params:- n: '14'- number: 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-- params:- n: '15'- number: 1/42849873690624000*x^14 - 1/1224282105446400*x^13 + 1/94175546572800*x^12+ '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-- params:- n: '16'- number: 1/1371195958099968000*x^15 - 1/34279898952499200*x^14 + 1/2260213117747200*x^13+ '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
- 10662539/34708397689405440000*x^3 - 5281/333734593167360000*x^2 + 52037/723091618529280000*x + 3617/170729965486080000-- params:- n: '17'- number: 1/46620662575398912000*x^16 - 1/1028396968574976000*x^15 + 1/58765541061427200*x^14+ '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
- 32891/1416669293445120000*x^4 + 25177/788827220213760000*x^3 + 52037/1446183237058560000*x^2 + 3617/341459930972160000*x-- params:- n: '18'- number: 1/1678343852714360832000*x^17 - 1/32908702994399232000*x^16 + 1/1645435149719961600*x^15+ '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
+ 530941/277667181515243520000*x^5 + 473129/51000094564024320000*x^4 + 9473/1388335907576217600*x^3 + 7132679/55759040888029839360000*x^2 - 1295681/774431123444858880000*x - 43867/91963695909076992000-- params:- n: '19'- number: 1/63777066403145711616000*x^18 - 1/1118895901809573888000*x^17 + 1/49363054491598848000*x^16+ '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
+ 170827/238000441298780160000*x^6 + 1314421/999601853454876672000*x^5 + 869/2103539253903360000*x^4 - 7023661/8578313982773821440000*x^3 - 1295681/1548862246889717760000*x^2 - 43867/183927391818153984000*x-- params:- n: '20'- number: 1/2551082656125828464640000*x^19 - 1/40280252465144659968000*x^18 + 1/1579617743731163136000*x^17+ '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
- 513310799/3345542453281790361600000*x^3 + 24989087/14125623691634225971200000*x^2 + 152388293/3884546515199412142080000*x + 174611/16057153253965824000000-- params:- n: '21'- number: 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-- params:- n: '22'- number: 1/4714400748520531002654720000*x^21 - 1/61225983747019883151360000*x^20- + 1/1933452118326943678464000*x^19 - 83/9667260591634718392320000*x^18 + 1/12924145175982243840000*x^17+ '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
+ 116927697319/33562481891322920907571200000*x^3 - 23163719/166480564937117663232000000*x^2 - 269839961/291340988639955910656000000*x - 77683/310224200866619719680000-- params:- n: '23'- number: 1/216862434431944426122117120000*x^22 - 1/2571491317374835092357120000*x^21+ '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
- 132947/254362971638071296000000*x^3 - 269839961/582681977279911821312000000*x^2 - 77683/620448401733239439360000*x-- params:- n: '24'- number: 1/10409396852733332453861621760000*x^23 - 1/113145617964492744063713280000*x^22+ '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
+ 576654122683/108071191690059805322379264000000*x^2 + 19311547193251/878078432481735918244331520000000*x + 236364091/40651779281561848066867200000-- params:- n: '25'- number: 1/520469842636666622693081088000000*x^24 - 1/5204698426366666226930810880000*x^23+ '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
Sign in to restore an earlier version.
