combinatorics – $sumlimits_{k=0}^{n} kbinom{2n}{k} = n2^{2n -1} overset{?}{iff}$ Vandermonde’s Identity?

Does $sumlimits_{k=0}^{r}{ {{m}choose{k}} { {n} choose {r-k} } } = {m + n choose r} overset{?}{iff} $ $sumlimits_{k=0}^{n} kbinom{2n}{k} = n2^{2n -1}$ ?

If I substitute $m = 2n$ into Vandermonde’s Identity, then $sumlimits_{k=0}^{r} {{m}choose{k}} {{n} choose {r-k}} = {m + n choose r} overset{m = 2n}{iff} sumlimits_{k=0}^{r} {{2n}choose{k}}{color{red}{{n} choose {r-k}}} = {3n choose r}$.

How can I transmogrify $color{red}{{n} choose {r-k}}$ into $k$? Does this transmogrification help me succeed?