According to sage your first sum is:
$$ \frac{8^{n} 2^{n} \left(n - \frac{1}{2}\right)!}{\sqrt{\pi} n!} $$
According to Maple your second sum is:
$${3\,n\choose 2\,n}{2F1(-n,-4\,n-M;\,-3\,n;\,-1)}-{4\,n+M\choose n+1}{ 2\,n-1\choose 2\,n}{3F2(1,1,-3\,n-M+1;\,n+2,-2\,n+1;\,-1)}$$
For $M=2$:
$$\frac{2 \, {\left(2 \, n \Gamma\left(\frac{3}{4}\right) \Gamma\left(\frac{1}{4}\right) + \Gamma\left(\frac{3}{4}\right) \Gamma\left(\frac{1}{4}\right)\right)} 8^{n} 2^{n} \left(n - \frac{1}{2}\right)!^{2} - \pi 4^{2 \, n + 1} \left(n + \frac{1}{4}\right)! \left(n - \frac{1}{4}\right)!}{\sqrt{\pi} \left(n + 1\right)! \left(n - \frac{1}{2}\right)! \Gamma\left(\frac{3}{4}\right) \Gamma\left(\frac{1}{4}\right)}
$$
For $M=3$:
$$
\frac{2 \, {\left(2 \, {\left(6 \, n^{2} \Gamma\left(\frac{3}{4}\right) \Gamma\left(\frac{1}{4}\right) + 7 \, n \Gamma\left(\frac{3}{4}\right) \Gamma\left(\frac{1}{4}\right) + 2 \, \Gamma\left(\frac{3}{4}\right) \Gamma\left(\frac{1}{4}\right)\right)} 8^{n} 2^{n} \left(n - \frac{1}{2}\right)!^{2} - \pi 4^{2 \, n + 2} \left(n + \frac{3}{4}\right)! \left(n + \frac{1}{4}\right)!\right)}}{\sqrt{\pi} \left(n + 2\right)! \left(n - \frac{1}{2}\right)! \Gamma\left(\frac{3}{4}\right) \Gamma\left(\frac{1}{4}\right)}
$$