\displaystyle{\sum_{i\ge 0}\sum_{j\ge 0} {2i+j-i\choose i}x^{2i+j}=\sum_{i\ge 0}x^{2i}\sum_{j\ge 0} {i+j\choose i}x^j=\sum_{i\ge 0}{x^{2i}\over (1-x)^{i+1}}=}