evaluating summation with counting formula

Question

I am trying to calculate what $\displaystyle\sum_{i=0}^{T}$$\displaystyle\sum_{j=0}^{P/2}\binom{i+j}{i}$ is equal to with some effort outside the scope of this article, and I'm having trouble with expressing the right way to do it. Please help

Answer 1

Using the Hockey-stick identity, we have, $\displaystyle\sum_{i=0}^{T}\binom{i+j}{i}=\sum_{i=0}^{T}\binom{i+j}{j}=\binom{j+T+1}{j+1}=\binom{j+T+1}{T}$.

Using the same identity again, we have, $\displaystyle\sum_{i=0}^{T}\sum_{j=0}^{\frac{P}{2}}\binom{i+j}{i}=\sum_{j=0}^{\frac{P}{2}}\binom{j+T+1}{T}=\binom{\frac{P}{2}+T+2}{T+1}-1$ .