  Let $c=(C_n)_{n\ge 0}$ be the Catalan sequence and $T$ a linear and bounded
operator on a Banach space $X$ such $4T$ is a power-bounded operator. The
Catalan generating function is defined by the following Taylor series, $$
C(T):=\sum_{n=0}^\infty C_nT^n. $$ Note that the operator $C(T)$ is a solution
of the quadratic equation $TY^2-Y+I=0.$ In this paper we define powers of the
Catalan generating function $C(T)$ in terms of the Catalan triangle numbers. We
obtain new formulae which involve Catalan triangle numbers; the spectrum of
$c^{\ast j}$ and the expression of $c^{-\ast j}$ for $j\ge 1$ in terms of
Catalan polynomials ($\ast$ is the usual convolution product in sequences). In
the last section, we give some particular examples to illustrate our results
and some ideas to continue this research in the future.
