Convolution of binomial rows

Vandermonde variant. Let $m=k-j$, so $\sum_{k}\binom{k}{j}\binom{n-k}{i}=\sum_m\binom{j+m}{j}\binom{i+n-j-m}{i}$.

Using the generating function $\sum_m\binom{j+m}{j}x^m=\frac{1}{(1-x)^{j+1}}$ and $\sum_m\binom{i+m}{i}x^m=\frac{1}{(1-x)^{i+1}}$, the convolution is $\frac{1}{(1-x)^{i+j+2}}$, whose coefficient is $\binom{n+1}{i+j+1}$.

Conclusion. $\sum_k\binom{k}{j}\binom{n-k}{i}=\binom{n+1}{i+j+1}$.