Partial Partial Sum of Even / Odd Binomial Coefficients

I have a following sum:

$ S_g = sum_ {k = 0} ^ gk binom {4g + 2} {2k} $

I can turn 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 for dealing with any of the above sums?