I have a polynomial that looks like
$$p_{3,10}(x_0, x_1, ... x_9) = (x_0 x_1 x_2 x_3 x_4 + x_0 x_1 x_2 x_3 x_5 + ... + x_5 x_6 x_7 x_8 x_9)^3$$
How do I determine the formula for the coefficient of each term in the expanded polynomial? Note that inside the parentheses, we are summing a total number of ${10}\choose{5}$ $x_{i}$'s.
In general, I would like a formula for the coefficient of each term of the more general polynomial
$$p_{r,t}(x_0, x_1, ... x_t) = (x_0 ... x_{t/2 - 1} + x_0 ... x_{t/2 - 2} x_{t/2} + ... + x_{t/2} ... x_{t - 1})^r$$
I have tried brute force algebraic manipulation. I have also tried using combinatorial arguments, but I can't quite make it work. I am now trying to find a recurrence relation in terms of $r$ for a fixed $t$, and I will try to find a recurrence relation in of $t$ for a fixed $r$ as well. If you have any other ideas I would be grateful!
Note: I forgot to mention the following two constraints: $r \in \mathbb{N}$, and $t \in \mathbb{N}$ such that $2 \vert t$.
Update
It was pointed out to me that this problem is equivalent to counting how many ways a certain $15$-element multiset can decompose into $3$ $5$-element sets. I am researching how to express this counting formula explicitly.
Second update
I found this paper which seems to offer a way of counting the sets as described in my previous update. Please let me know if anyone has advice on how to adapt the formula on page 6 to my current problem.