We set, for all $t \in I$,
$$f ( t ) = \sum _ { n = 0 } ^ { + \infty } C _ { n } t ^ { n } \quad \text { and } \quad g ( t ) = 2 t f ( t ) .$$
The generating series of $T$ is given by $G _ { T } ( t ) = \sum _ { n = 0 } ^ { \infty } \mathbb { P } ( T = n ) t ^ { n }$, defined if $t \in [ - 1,1 ]$. Using the previous questions, express $G _ { T }$ using $g$ and $\mathbb { P } ( T = 0 )$.