For the Motzkin and Schröder numbers respectively, we have the following identities: $$ Mk(z) = \sum_{n=1}^{\infty} \Bigg{(} -\frac{1}{2} \sum_{a=0}^{n+2} (-3)^{k} \binom{\frac{1}{2}}{a} \binom{ \frac{1}{2}}{b} \Bigg{)}z^{n} = \frac{1 - z - \sqrt{1-2z-3z^2}}{2z^2} \quad ,$$ (where: $b = n + 2 - a $) and $$S(z) = \sum_{n=1}^{\infty} s_{n} z^{n} = \frac{1-x-\sqrt{1-6z+z^2}}{2z} \quad , $$ where $s_{n}$ is given by $$ s_{n} = s_{n-1} + \sum_{k=0}^{n-1}s_{k}s_{n-1-k} \quad . $$ I obtained these identities from these pages on mathworld. However, I can't find where the author(s) of the mathworld article found the aforementioned identities within the list of references he or she provides.
Can you please help me find a correct reference for both of these formulas?
These are just a few with the generating functions or expansions.
1) Matthias Schork 2) Robert A. Sulanke 3) Sen-Peng Eu 4) Toufik Mansour 5) Ira M. Gessel 6) Eva Y. P. Deng