Quote by AtomSeven
The identity is wrong, it should be
[tex]
\sum_{i=0}^{n}2^{ni} {n+i \choose i}=2^{2 n}
[/tex]

Version of AtomSeven is good.
How can I show that [tex]
\sum_{i=0}^{n}2^{ni} {n+i \choose i}=2^{2 n}
[/tex]
is good for every natural numbers