Explicit formulas for the Dumont-Foata polynomial

Dumont and Foata have defined a polynomial F"n(x, y, z) recursively. They proved that F"n(x, y, z) is symmetric in x, y, z and that F"n(1, 1, 1) = G"2"n"+"2 the Genocchi number. Moreover, they gave an elegant combinatorial interpretation for the coefficients of F"n(x, y, z). In the present paper explicit formulas and generating functions for F"n(x, y, z) are obtained.