Other summation properties of the Eulerian numbers are:

The Eulerian numbers of the second kind satisfy the recurrence relation, that follows directly from the above definition:

There are formulas connecting Eulerian numbers to Bernoulli numbers:

Worpitzky's identity expresses x as the linear combination of Eulerian numbers with binomial coefficients:

In addition to working on several problems of probability which link to combinatorics, he worked on the knights tour, Graeco-Latin square, Eulerian numbers, and others.

An alternative summation formula expresses the ordered Bell numbers in terms of the Eulerian numbers, which count the number of permutations of n items with k + 1 runs of increasing items:

Triangular arrays of integers in which each row is symmetric and begins and ends with 1 are sometimes called generalized Pascal triangles; examples include Pascal's triangle, the Narayana numbers, and the triangle of Eulerian numbers.