Suppose, I am having difficulties writing down as double summation. The expression is $$b_1(t_n)\sum\limits_{k = 1}^1 {a_{1k}u_k}+b_2(t_n)\sum\limits_{k = 1}^2 {a_{2k}u_k}+ ... +b_p(t_n)\sum\limits_{k = 1}^p {a_{pk}u_k}$$
It's probably pretty simple(with a less than/condition in a summation) but can't come up with the correct one.
Here we see that the upper indices $1,\ldots,p$ of the sums correspond to the indices $j, 1 \leq j \leq p$ of the factors $b_j\left(t_n\right)$ and $a_{jk}$. We can therefore write \begin{align*} b_1(t_n)\sum_{k = 1}^1 {a_{1k}u_k}+b_2(t_n)\sum_{k = 1}^2 {a_{2k}u_k}+ \cdots+b_p(t_n)\sum_{k = 1}^p {a_{pk}u_k} =\sum_{\color{blue}{j}=1}^pb_\color{blue}{j}(t_n)\sum_{k = 1}^{\color{blue}{j}} {a_{\color{blue}{j}k}u_k} \end{align*}