  We establish a simple recurrence formula for the number $Q_g^n$ of rooted
orientable maps counted by edges and genus. We also give a weighted variant for
the generating polynomial $Q_g^n(x)$ where $x$ is a parameter taking the number
of faces of the map into account, or equivalently a simple recurrence formula
for the refined numbers $M_g^{i,j}$ that count maps by genus, vertices, and
faces. These formulas give by far the fastest known way of computing these
numbers, or the fixed-genus generating functions, especially for large $g$. In
the very particular case of one-face maps, we recover the Harer-Zagier
recurrence formula.
  Our main formula is a consequence of the KP equation for the generating
function of bipartite maps, coupled with a Tutte equation, and it was
apparently unnoticed before. It is similar in look to the one discovered by
Goulden and Jackson for triangulations, and indeed our method to go from the KP
equation to the recurrence formula can be seen as a combinatorial
simplification of Goulden and Jackson's approach (together with one additional
combinatorial trick). All these formulas have a very combinatorial flavour, but
finding a bijective interpretation is currently unsolved.
