Given the following set of equations from $t \in [1,T]$ $$X^1 = Nq$$ $$X^2 = 2Nq - q^2[N_1]$$ $$X^3 = 3Nq - q^2[2N_1 + N_2] + q^3[N_2]$$ $$X^4 = 4Nq - q^2[3N_1 + 2N_2 + N_3] + q^3[2N_2+2N_3] - q^4[N_4]$$ $$X^5 = 5Nq - q^2[4N_1 + 3N_2 + 2N_3 + N_4] + q^3[3N_2 + 4N_3 + 3N_4] - q^4[2N_3 + 2N_4] + q^5[N_5]$$ $$X^6 = 6Nq - q^2[5N_1 + 4N_2 ... + N_5] + q^3[4N_2+6N_3+6N_4+4N_5] - q^4[...] + q^5[...] - q^6[N_6]$$ $$X^7 = 7Nq - q^2[6N_1 ... + N_6] + q^3[N_3] - q^4[...] + q^5[...] - q^6[...] + q^7[...]$$ $$...$$ I can formulate the above as follows, considering that the brackets are all following the same pattern, however, note that the values in square brackets are obtained through an algorithm. I don't know how to put it here. In the following, I have used $A_t$. Please feel free to change it to any form.
$$X^T = TNq + \sum^T_{t=2} (-1)^{n+1}q^{t}A_t$$
where $A_t$ is different in all the cases and is not a constant but depends on my experiment. However, the sum of all the coefficients for any power of $q$ i.e. $q^t$ can be found as $\binom{T}{t}$. (EDIT: Can we use matrix formulation???)
Hope I am able to clarify my problem.