  Springer numbers are an analog of Euler numbers for the group of signed
permutations. Arnol'd showed that they count some objects called snakes, that
generalize alternating permutations. Hoffman established a link between
Springer numbers, snakes, and some polynomials related with the successive
derivatives of trigonometric functions.
  The goal of this article is to give further combinatorial properties of
derivative polynomials, in terms of snakes and other objects: cycle-alternating
permutations, weighted Dyck or Motzkin paths, increasing trees and forests. We
obtain the generating functions, in terms of trigonometric functions for
exponential ones and in terms of J-fractions for ordinary ones. We also define
natural q-analogs, make a link with normal ordering problems and combinatorial
theory of differential equations.

Combinatorics