I have a following sum:


$S\_g=\sum\_{k=0}^g k\binom{4g+2}{2k}$


I can transform it into a different sum


$S\_g=(2g+1)\sum\_{k=1}^g\binom{4g+1}{2k-1}$


What is the closed form or what is the method to deal with any of above sums?



Specific partial sum of even/odd binomial coefficients
According to Maple,
$$ S\_g = \left( g + \frac12\right) \left(16^g - {4 g \choose 2g}\right) $$


