The enumeration of rooted maps has been first studied by W. T. Tutte in the early 1960's, with planar maps. New results have been obtained since for rooted maps on more general surfaces (torus with 1, 2, 3 holes, projective plane,...). I present the enumeration of rooted maps on the Klein bottle as a first step to the general case. Then I give an other approach of the enumeration of rooted maps, the rooted maps regardless to the genus of their associated surface. This leads to Riccati equations whose solutions are expressed as continued fractions. I obtain also a new equation generalizing the Dyck equation for rooted planar maps.
P_{0}(u,z) 

(1)  
P_{1}(u,z) 

(2)  
P_{2}(u,z) 

(3)  

P_{2}(u,z)A(u,z)=uzP_{2}(1,z)+(1u)u^{2}z 
æ ç ç è 

[  uP_{1}(u,z)  ]  +P_{1}(u,z)^{2}+P_{0}(u,u,z) 
ö ÷ ÷ ø 
. 




(4) 
M(y,z)= 

T(z)= 
é ê ê ë 

ù ú ú û 

. 
T(z))=1+zT(z)+2z^{2} 

T(z). 
T(z)= 

. 
T(z)= 

. 
This document was translated from L^{A}T_{E}X by H^{E}V^{E}A.