Several numbers are named after Euler.
Euler’s number is the irrational number typically denoted “e” and defined by the series
This happens to also be the groupoid cardinality of the groupoid , the core of FinSet.
The Euler numbers (also called secant numbers) , are a sequence of integers defined via the generating function
All odd-numbered members of the sequence vanish: for . , , , see Euler number at wikipedia for more and the Abramowitz–Stegun handbook for many Euler numbers.
Euler numbers generalize to Euler polynomials defined via the generating function
so that . The Euler polynomials satisfy the recursion and may be conversely expressed via Euler numbers as
There is also a complementarity formula . See e.g. Louis Comtet, Advanced combinatorics, D. Reifel Publ. Co., Dordrecht-Holland, Boston 1974 djvu
There are also Eulerian numbers (forming a different, double sequence ). Combinatorially, counts the number of permutations of the set with ascents (an ascent of is an element such that ). Alternatively, the double sequence can be defined recursively through the formula
There is also a number, usually denoted as , called the Euler-Mascheroni constant. It is defined as a limit
where is the natural logarithm. It arises for example in discussions of the Gamma function.