Rising Sum of Binomial Coefficients/Marginal Cases

Theorem
Let $n \in \Z$ be an integer such that $n \ge 0$.

Then:


 * $\displaystyle \sum_{j \mathop = 0}^m \binom {n + j} n = \binom {n+m+1} {n+1} = \binom {n+m+1} m$

where $\displaystyle \binom n k$ denotes a binomial coefficient.

n = 0
When $n = 0$ we have:

So the theorem holds for $n = 0$.

n = 1
When $n = 1$ we have:

So the theorem holds for $n = 1$.