I can write a general term, say $m$th term $T_m$, as a sum of $2^{m-1}$ terms in which each term contains two type of variables, i.e., $a_{i,j}$ and $b_i$ where $i,j\in 0,\cdots,m$. However, I do not see any pattern to write it in compact form with sum and product notations. But I know that each term at least has one $a_{i,j}$ and $b_i$ product at most has product with $m$ number of $a_{i,j}$ and $m$ number of $b_i$.
E.g., the 4th term, i.e., $m=4$ so I have $2^3=8$ terms where $T_4$ is: $$T_4=a_{0,4}b_0 +a_{0,3}a_{3,4}b_0b_4+a_{0,1}a_{1,4}b_0b_1 +a_{0,2}a_{2,4}b_0b_0 +a_{0,1}a_{1,2}a_{2,4}b_0b_1b_2 +a_{0,1}a_{1,3}a_{3,4}b_0b_1b_4 +a_{0,2}a_{2,3}a_{3,4}b_0b_3b_4 +a_{0,1}a_{1,2}a_{2,3}a_{3,4}b_0b_1b_3b_4$$
I can notice that number of $a_{i,j}$s and $b_i$s are equal for a given term.
Can someone help me to write $T_m$ expression in general way, $\sum_{i,j}^{}\prod_{i,j}^{}a_{i,j}b_i$, which may not give the exact expression but to represent the rough pattern?
By including the fact that each term at least has one $a_{i,j}$ and $b_i$ product (e.g., $a_{0,4}b_0$), and at most has product with $m$ number of $a_{i,j}$ and $m$ number of $b_i$ (e.g., $a_{0,1}a_{1,2}a_{2,3}a_{3,4}b_0b_1b_3b_4$).