Let $B_n$ be the $n$-th Catalan Number. We have $ B(x) = \sum_{n \ge 0} B_n x^n = \frac{1-\sqrt{1-4x}}{2x}$.
Does anyone know a closed form of the generating function of the shifted Catalan Numbers, i.e. for chosen $p_0$, for $B_{p_0}(x) = \sum_{n \ge 0} B_{n+p_0} x^n$?
If I understand the question correctly, this isn't too difficult nor is it particular to Catalan numbers in any way. Let $f(x)$ be a generating function for the sequence $\{a_n\}_{n=0}^\infty$. Define the polynomial
$$P_m(x)=\sum_{n=0}^{m-1} a_n x^n.$$
(For $m=0$ we say $P\equiv0$.) This is just the series expansion of $f(x)$ truncated. Then we have
$$\sum_{n\ge0} a_{n+m}x^n=x^{-m}\left(\sum_{n\ge0}a_{n+m}x^{n+m}\right)=\frac{f(x)-P_m(x)}{x^m}.$$
Is this what you're looking for or were you interested in something different?