Peter Luschny, 2010−08−18
Contents 
The Eulerian polynomials are defined by the exponential generating function
The Eulerian polynomials can be computed by recurrence:
An equivalent way to write this definition is to set the Eulerian polynomials inductively by
The definition given is used by major authors like D. E. Knuth, D. Foata and F. Hirzebruch. In the older literature (for example in L. Comtet, Advanced Combinatorics) a slightly different definition is used, namely
The coefficients of the Eulerian polynomials are the Eulerian numbers A_{n,k} ^{[1]},
This definition of the Eulerian numbers agrees with the combinatorial definition in the DLMF ^{[2]}. The triangle of Eulerian numbers is also called Euler's triangle^{ [3]}.


Euler's definition A_{n,k} is A173018. The main entry for the Eulerian numbers in the database is A008292. It enumerates C_{n,k} with offset (1,1).
Let S_{n} denote the set of all bijections (onetoone and onto functions) from {1, 2, …, n} to itself, call an element of S_{n} a permutation p and identify it with the ordered list p_{1} p_{2} … p_{n}.
Using the Iverson bracket [.] the number of ascents of p is defined as
where p_{n+1} ← 0. The combinatorial interpretation of the Eulerian polynomials is then given by
The table below illustrates this representation for the case n = 4.
p  asc  p  asc  p  asc  p  asc 
4321  0  4231  1  2413  2  1423  2 
3214  1  2431  1  2134  2  1342  2 
3241  1  4312  1  2314  2  4123  2 
3421  1  3142  1  2341  2  1324  2 
4213  1  4132  1  3124  2  1243  2 
2143  1  1432  1  3412  2  1234  3 
The number of permutations of {1, 2, …, n} with n ascents (the central Eulerian numbers) are listed in A180056.
Eulerian polynomials in Institutiones calculi differentialis, 1755
Leonhard Euler introduced the polynomials in 1749 ^{ [4]} in the form
Euler introduced the Eulerian polynomials in an attempt to evaluate the Dirichlet eta function
at s = 1, 2, 3,... . This led him to conjecture the functional equation of the eta function (which immediately implies the functional equation of the zeta function). Most simply put, the relation Euler was after was
Though Euler's reasoning was not rigorous by modern standards it was a milestone on the way to Riemann's proof of the functional equation of the zeta function.
The facsimile shows Eulerian polynomials as given by Euler in his work Institutiones calculi differentialis, 1755. It is interesting to note that the original definition of Euler coincides with the definition in the DLMF, 2010.
We call a generating function an Eulerian generating function iff it has the form
for some polynomial g(t). Many elementary classes of sequences have an Eulerian generating function. A few examples are collocated in the table below.
Generating function g(t)A_{n}(t)/(1t)^{n+1} 
n = 0  n = 1  n = 2  n = 3  n = 4  n = 5 
g(t) = 1 − t^{2}  A019590  A040000  A008574  A005897  A008511  A008512 
g(t) = 1 − t  A000007  A000012  A005408  A003215  A005917  A022521 
g(t) = t  A057427  A001477  A000290  A000578  A000583  A000584 
g(t) = 1 + t  A040000  A005408  A001844  A005898  A008514  A008515 
g(t) = 1 + t + t^{2}  A158799  A008486  A005918  A027602  A160827  A179995 
For instance the case
A_{1}(x) , A_{2}(x) , A_{3}(x) , A_{4}(x) , A_{5}(x) , A_{6}(x) 
x  −1/2  1/2  3/2 
2^{n}A_{n}(x)  A179929  A000629  A004123 
x  −2  −1  0 
A_{n}(x)  A087674  A155585  A000012 
x  1  2  3 
A_{n}(x)  A000142  A000670  A122704 
Let ∂r denote the denominator of a rational number r.
A122778  A_{n}(n) 
A180085  A_{n}(−n) 
A006519  ∂(A_{n}(−1) / 2^{n}) 
A001511  log_{2}(∂(A_{2n+1}(−1) / 2^{2n+1})) 
Eulerian polynomials A_{n}(x) and Euler polynomials E_{n}(x) have a sequence of values in common (up to a binary shift). Let B_{n}(x) denote the Bernoulli polynomials and ζ(n) the Riemann Zeta function. denotes the Stirling numbers of the second kind. The formulas below show how rich in content the Eulerian polynomials are.
A155585 for all n ≥ 0 
Eulerian polynomials are related to the polylogarithm
For nonpositive integer values of s, the polylogarithm is a rational function. The first few are
A plot of these functions in the complex plane is given in the gallery^{ [5]} below.
In general the explicit formula for nonpositive integer s is
See also DLMF and the section on series representations of the polylogarithm on Wikipedia. However, note that the conventions on Wikipedia do not conform to the DLMF definition of the Eulerian polynomials.
This article was originally written for the OeisWiki. Thanks to Daniel Forgues for editorial help. It is also available in pdf format.