I have a series of equations for the coefficients of a polynom. For every order of the coefficient there are more terms. I'm trying to write this in a closed form for any coefficient, but I'm stuck on how to do this. Here is what I have so far:
$$ s^1: \sum_{i = 1}^{N} \sum_{j = i}^{N} R_i \cdot C_j$$
$$ s^2: \sum_{i = 1}^{N - 1} \sum_{j = i}^{N} \sum_{k = i}^{j - 1} \sum_{l = j}^{N} R_i \cdot R_j \cdot C_k \cdot C_l $$
$$ s^3: \sum_{i = 1}^{N - 1} \sum_{j = i + 1}^{N - 1} \sum_{k = j + 1}^{N} \sum_{l = i}^{j - 1} \sum_{m = j}^{k - 1} \sum_{n = k}^{N} R_i \cdot R_j \cdot R_k \cdot C_l \cdot C_m \cdot C_n $$ $$ s^4: \sum_{i = 1}^{N - 1} \sum_{j = i + 1}^{N - 1} \sum_{k = j + 1}^{N} \sum_{l = k + 1}^{N} \sum_{m = i}^{j - 1} \sum_{n = j}^{k - 1} \sum_{o = k}^{l - 1} \sum_{p = l}^{N} R_i \cdot R_j \cdot R_k \cdot R_l \cdot C_m \cdot C_n \cdot C_o \cdot C_p $$
And so on. I think I've figured out the structure for $s^n$, as I tried this up sixth-order ($s^6$), which works out (I can check this against other representations). What I would like to have is a form where the structure (number of summation symbols and R's and C's) does not change depending on the order of the coefficient:
$$s^i = ?$$
Are there standard techniques or some practical tips on how to tackle this?
Some background: I'm trying to derive a closed form of the transfer function of a cascaded RC-network (electronics). The equations describe the coefficients of the polynom in the denominator of the transfer function. Ultimately, I need to implement this in a program. As I can't hard-code the number of for-loops, I'm searching for a more suitable representation, perhaps also a representation that lends itself for a recursive implementation. Any guidance is welcome.