  In this paper we determine possible decompositions of Euler polynomials
$E_k(x)$, i.e. possible ways of writing Euler polynomials as a functional
composition of polynomials of lower degree. Using this result together with the
well-known criterion of Bilu and Tichy, we prove that the Diophantine equation
$$-1^k +2 ^k - \cdots + (-1)^{x} x^k=g(y),$$ with $g\in \mathbb{Q}[X]$ of
degree at least $2$ and $k\geq 7$, has only finitely many integers solutions
$x, y$ unless polynomial $g$ can be decomposed in ways that we list explicitly.
