The Mathematica command
Sum[k*Binomial[4 g + 2, 2 k], {k, 0, g}]//FullSimplify
performs $$-\binom{4 g+2}{2 (g+2)} \, _3F_2\left(2,1-g,\frac{3}{2}-g;g+\frac{5}{2},g+3;1\right)-\frac{(g+1) \binom{4 g+2}{2 (g+1)} (2)_{g-\frac{1}{2}} \left(g+\frac{3}{2}\right)_{g-\frac{1}{2}}}{\left(\frac{3}{2}\right)_{g-\frac{1}{2}} (g+2)_{g-\frac{1}{2}}}+16^g (2 g+1) .$$$$16^g g-\frac{\Gamma (4 g+3) \, _3F_2\left(2,1-g,\frac{3}{2}-g;g+\frac{5}{2},g+3;1\right)}{\Gamma (2 g-1) \Gamma (2 g+5)}.$$