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Abel polynomials $A_n(x;a)$
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Numbers
$n$ 
0:
1
1:
x
2:
x^2 - 2*x*a
3:
x^3 - 6*x^2*a + 9*x*a^2
4:
x^4 - 12*x^3*a + 48*x^2*a^2 - 64*x*a^3
5:
x^5 - 20*x^4*a + 150*x^3*a^2 - 500*x^2*a^3 + 625*x*a^4
6:
x^6 - 30*x^5*a + 360*x^4*a^2 - 2160*x^3*a^3 + 6480*x^2*a^4 - 7776*x*a^5
7:
x^7 - 42*x^6*a + 735*x^5*a^2 - 6860*x^4*a^3 + 36015*x^3*a^4 - 100842*x^2*a^5 + 117649*x*a^6
8:
x^8 - 56*x^7*a + 1344*x^6*a^2 - 17920*x^5*a^3 + 143360*x^4*a^4 - 688128*x^3*a^5 + 1835008*x^2*a^6 - 2097152*x*a^7
9:
x^9 - 72*x^8*a + 2268*x^7*a^2 - 40824*x^6*a^3 + 459270*x^5*a^4 - 3306744*x^4*a^5 + 14880348*x^3*a^6 - 38263752*x^2*a^7 + 43046721*x*a^8
10:
x^10 - 90*x^9*a + 3600*x^8*a^2 - 84000*x^7*a^3 + 1260000*x^6*a^4 - 12600000*x^5*a^5 + 84000000*x^4*a^6 - 360000000*x^3*a^7 + 900000000*x^2*a^8 - 1000000000*x*a^9
11:
x^11 - 110*x^10*a + 5445*x^9*a^2 - 159720*x^8*a^3 + 3074610*x^7*a^4 - 40584852*x^6*a^5 + 372027810*x^5*a^6 - 2338460520*x^4*a^7 + 9646149645*x^3*a^8 - 23579476910*x^2*a^9 + 25937424601*x*a^10
12:
x^12 - 132*x^11*a + 7920*x^10*a^2 - 285120*x^9*a^3 + 6842880*x^8*a^4 - 114960384*x^7*a^5 + 1379524608*x^6*a^6 - 11824496640*x^5*a^7 + 70946979840*x^4*a^8 - 283787919360*x^3*a^9 + 681091006464*x^2*a^10 - 743008370688*x*a^11
13:
x^13 - 156*x^12*a + 11154*x^11*a^2 - 483340*x^10*a^3 + 14137695*x^9*a^4 - 294064056*x^8*a^5 + 4459971516*x^7*a^6 - 49696825464*x^6*a^7 + 403786706895*x^5*a^8 - 2332989862060*x^4*a^9 + 9098660462034*x^3*a^10 - 21505924728444*x^2*a^11 + 23298085122481*x*a^12
14:
x^14 - 182*x^13*a + 15288*x^12*a^2 - 784784*x^11*a^3 + 27467440*x^10*a^4 - 692179488*x^9*a^5 + 12920683776*x^8*a^6 - 180889572864*x^7*a^7 + 1899340515072*x^6*a^8 - 14772648450560*x^5*a^9 + 82726831323136*x^4*a^10 - 315866083233792*x^3*a^11 + 737020860878848*x^2*a^12 - 793714773254144*x*a^13
15:
x^15 - 210*x^14*a + 20475*x^13*a^2 - 1228500*x^12*a^3 + 50675625*x^11*a^4 - 1520268750*x^10*a^5 + 34206046875*x^9*a^6 - 586389375000*x^8*a^7 + 7696360546875*x^7*a^8 - 76963605468750*x^6*a^9 + 577227041015625*x^5*a^10 - 3148511132812500*x^4*a^11 + 11806916748046875*x^3*a^12 - 27246730957031250*x^2*a^13 + 29192926025390625*x*a^14
16:
x^16 - 240*x^15*a + 26880*x^14*a^2 - 1863680*x^13*a^3 + 89456640*x^12*a^4 - 3148873728*x^11*a^5 + 83969966080*x^10*a^6 - 1727382159360*x^9*a^7 + 27638114549760*x^8*a^8 - 343940981063680*x^7*a^9 + 3301833418211328*x^6*a^10 - 24013333950627840*x^5*a^11 + 128071114403348480*x^4*a^12 - 472877960873902080*x^3*a^13 + 1080863910568919040*x^2*a^14 - 1152921504606846976*x*a^15
17:
x^17 - 272*x^16*a + 34680*x^15*a^2 - 2751280*x^14*a^3 + 152008220*x^13*a^4 - 6201935376*x^12*a^5 + 193293652552*x^11*a^6 - 4694274419120*x^10*a^7 + 89777998265670*x^9*a^8 - 1356645307125680*x^8*a^9 + 16144079154795592*x^7*a^10 - 149699643071740944*x^6*a^11 + 1060372471758165020*x^5*a^12 - 5546563698427324720*x^4*a^13 + 20205339187128111480*x^3*a^14 - 45798768824157052688*x^2*a^15 + 48661191875666868481*x*a^16
18:
x^18 - 306*x^17*a + 44064*x^16*a^2 - 3965760*x^15*a^3 + 249842880*x^14*a^4 - 11692646784*x^13*a^5 + 420935284224*x^12*a^6 - 11906455182336*x^11*a^7 + 267895241602560*x^10*a^8 - 4822114348846080*x^9*a^9 + 69438446623383552*x^8*a^10 - 795385843140575232*x^7*a^11 + 7158472588265177088*x^6*a^12 - 49558656380297379840*x^5*a^13 + 254873089955815096320*x^4*a^14 - 917543123840934346752*x^3*a^15 + 2064472028642102280192*x^2*a^16 - 2185911559738696531968*x*a^17
19:
x^19 - 342*x^18*a + 55233*x^17*a^2 - 5596944*x^16*a^3 + 398782260*x^15*a^4 - 21215216232*x^14*a^5 + 873359734884*x^13*a^6 - 28446574221936*x^12*a^7 + 743166751548078*x^11*a^8 - 15689075866014980*x^10*a^9 + 268283197308856158*x^9*a^10 - 3707185999176921456*x^8*a^11 + 41087978157544212804*x^7*a^12 - 360309962304618481512*x^6*a^13 + 2444960458495625410260*x^5*a^14 - 12387799656377835411984*x^4*a^15 + 44131536275846038655193*x^3*a^16 - 98646963440126439346902*x^2*a^17 + 104127350297911241532841*x*a^18
20:
x^20 - 380*x^19*a + 68400*x^18*a^2 - 7752000*x^17*a^3 + 620160000*x^16*a^4 - 37209600000*x^15*a^5 + 1736448000000*x^14*a^6 - 64496640000000*x^13*a^7 + 1934899200000000*x^12*a^8 - 47297536000000000*x^11*a^9 + 945950720000000000*x^10*a^10 - 15479193600000000000*x^9*a^11 + 206389248000000000000*x^8*a^12 - 2222653440000000000000*x^7*a^13 + 19051315200000000000000*x^6*a^14 - 127008768000000000000000*x^5*a^15 + 635043840000000000000000*x^4*a^16 - 2241331200000000000000000*x^3*a^17 + 4980736000000000000000000*x^2*a^18 - 5242880000000000000000000*x*a^19
Definition
The Abel polynomials are $A_0(x;a) = 1$ and $A_n(x;a) = x\,(x - a n)^{n-1}$ for $n \geq 1$. They are polynomials in two variables, $x$ and the parameter $a$.
Parameters
$n$
—   integer ($0 \leq n \leq 20$)
Formulas
(1)
$A_n(x;a) = x\,(x - a n)^{n-1}$ for $n \geq 1$, and $A_0 = 1$.
(2)
$A_n(x+y; a) = \sum_{k=0}^{n} \binom{n}{k} A_k(x;a)\, A_{n-k}(y;a)$: the Abel polynomials are of binomial type, which is the property that characterises them among Sheffer sequences.
(3)
$A_n(x;0) = x^{n}$.
(4)
$A_n(1;-1) = (n+1)^{n-1}$, the number of labelled trees on $n+1$ vertices (Cayley's formula). More generally $A_n(x;-1) = x(x+n)^{n-1}$ counts labelled rooted forests.
(5)
$A_n(x;a)$ is the image of $x^{n}$ under the umbral operator inverse to $f \mapsto f(x+a)$, which is what "Abel sequence" means.
Comments
(6)
Stored as polynomials in $x$ and $a$ together, rather than as a family indexed by $a$: $a$ is a parameter of the polynomial, not of the table. The same choice as the Dickson polynomials Dickson_polynomials_of_the_first_kind.
(7)
Twenty is where the longest entry reaches 551 characters, which is about where the polynomial tables here stop being something a person reads.
Programs
(P1)
Sage
R. = ZZ[]
def abel(n):
    return R(1) if n == 0 else x * (x - a*n)^(n - 1)

abel(21)                         # the next one after this table
Links
Data properties
Entries are of type: integral polynomial
Table is complete: false
How well the digits are known: exact


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