Question 1045495
It is enough to show that 

{{{(matrix(2,1,2n,0))+(matrix(2,1,2n,1))+(matrix(2,1,2n,2))}}}+...+ {{{(matrix(2,1,2n,2n)) = 4^n}}}.  (Why?)


It is known that {{{(1+x)^(2n) =1+(matrix(2,1,2n,1))*x+(matrix(2,1,2n,2))*x^2}}}+...+ {{{(matrix(2,1,2n,2n))*x^(2n)}}}.

Letting x = 1, get directly

{{{(1+1)^(2n) = 2^(2n) = 4^n = 1+(matrix(2,1,2n,1))+(matrix(2,1,2n,2))}}}+...+ {{{(matrix(2,1,2n,2n))}}}.

And that's it...