For $m=1$ we have $\binom{k_2}{k_1}$;
For $m=2$ we have $\binom{k_3}{k_2}\binom{k_2}{k_1}$;
For $m=3$ we have $\binom{k_4}{k_3}\binom{k_3}{k_2}\binom{k_2}{k_1}$. We continue this way.
I want to write for each $m$ the following way:
$\textbf{Example}:$ How can we write the following statement under the sum and multiplication symbol? We can also use sub-indices if necessary.
$$\binom{k_2}{k_1}+\binom{k_3}{k_2}\binom{k_2}{k_1}+\binom{k_4}{k_3}\binom{k_3}{k_2}\binom{k_2}{k_1}=\sum\displaystyle \prod_{i=1}^{m}\binom{k_{i+1}}{k_i}$$
I couldn't index the summation symbol.
It's $\sum_{m = 1}^3$: $$ \binom{k_2}{k_1}+\binom{k_3}{k_2}\binom{k_2}{k_1}+\binom{k_4}{k_3}\binom{k_3}{k_2}\binom{k_2}{k_1}=\sum_{m = 1}^3 \prod_{i=1}^{m}\binom{k_{i+1}}{k_i} $$ You can see this by inserting $m = 1, 2$ or $3$ into $\prod_{i=1}^{m}\binom{k_{i+1}}{k_i}$ and writing out what you get. For instance, inserting $m = 2$ gives $$ \prod_{i=1}^{2}\binom{k_{i+1}}{k_i} = \binom{k_2}{k_1}\binom{k_3}{k_2} $$ which is the second term from the sum above. Similarily, $m = 1$ gives the first term, and $m = 3$ gives the third term.