组合
\[\begin{aligned}
\sum_{j={l_2}}^{r_2}\sum_{i={l_1}}^{r_1}\binom{i+j}{j}&=\sum_{j={l_2}}^{r_2}\sum_{i=l_1+j}^{r_1+j}\binom{i}{j}\\
&=\sum_{j={l_2}}^{r_2}(\binom{r_1+j+1}{j+1}-\binom{l_1+j}{j+1})\\
&=\sum_{j=0}^{r_2}(\binom{r_1+j+1}{j+1}-\binom{l_1+j}{j+1})-\sum_{j=0}^{l_2-1}(\binom{r_1+j+1}{j+1}-\binom{l_1+j}{j+1})\\
&=\sum_{j=0}^{r_2}\binom{r_1+j+1}{j+1}-\sum_{j=0}^{r_2}\binom{l_1+j}{j+1}-\sum_{j=0}^{l_2-1}\binom{r_1+j+1}{j+1}+\sum_{j=0}^{l_2-1}\binom{l_1+j}{j+1}\\
&=\sum_{j=1}^{r_2+1}\binom{r_1+j}{j}-\sum_{j=1}^{r_2+1}\binom{l_1+j}{j}-\sum_{j=1}^{l_2}\binom{r_1+j}{j}+\sum_{j=1}^{l_2}\binom{l_1+j}{j}\\
&=\binom{r_1+r_2+2}{r_2+1}-\binom{l_1+r_2+2}{r_2+1}-\binom{r_1+l_2+1}{l_2}+\binom{l_1+l_2+1}{l_2}
\end{aligned}
\]
\[\begin{aligned} \sum_{i=l_1}^{r_1}\sum_{j=l_2}^{r_2}\binom{i+j}{i}&=\sum_{j=l_2}^{r_2}\sum_{i=l_1}^{r_1}\binom{i+j}{i}
\\
&=\sum_{j=l_2}^{r_2}(\binom{r_1+j+1}{r_1}-\binom{l_1+j}{l_1-1})
\\
&=\sum_{j=l_2}^{r_2}\binom{r_1+j+1}{r_1}-\sum_{j=l_2}^{r_2}\binom{l_1+j}{l_1-1}
\\
&=\sum_{i=l_2+r_1+1}^{r_2+r_1+1}\binom{i}{r_1}-\sum_{i=l_2+l_1}^{r_2+l_1}\binom{i}{l_1-1}
\\
&=\binom{r_1+r_2+2}{r_1+1}-\binom{r_1+l_2+1}{r_1+1}+\binom{l_1+l_2}{l_1}-\binom{l_1+r_2+1}{l_1}\end{aligned}
\]
