I am struggling to put a Series in a general form and was wondering if someone here could give a hand with that. If the question is to general or not meeting the standards, I apologize in advance.
Here it goes the example when T=4:
$a_4 = L_{0,4}(a_0 + s_1) + L_{1,3}s_2 + L_{2,2}s_{3} + W_{4}s_4$
What I need here is to find a general form, perhaps in summation form, for $a_{T}$.
Notice that:
1) The entries below every $L$ add up to $T$ but they are evolving in this fashion in which, from term to term in the summation, one has to take away $1$ from the right-hand side subscript and add up to the left-hand side subscript up to a point in which the last $L$ ($L_{2,2}$ in my example above) is in the form $L_{\frac{T}{2},\frac{T}{2}}$, For odd indexes, like $a_5$ I guess the only requirement is that the sum of the left-hand side subscript and the right one has to add up to $T$. (In this sense I guess that the left hand side subscript will be one unity bigger than the right-hand side one in the last term)
2) The last term is going to be invariably $W_Ts_T$
3) The term in parentheses in the first term always gonna be $(a_0 + s_1)$
Thank you very much!
Rewriting $a_4$ as follows may make it clearer: $$\;\;\ L_{0,4}a_0\\ \color{red}{+L_{0,4}s_1\\ +L_{1,3}s_2\\ +L_{2,2}s_3}\\ +W_4 s_4$$ which can then be written as $$\;\;\ L_{0,4}a_0\\ \color{red}{+\sum_{r=0}^2 L_{r,4-r}s_{r+1}}\\ +W_4 s_4$$
Substituting $4$ with $T$ and $2$ with $\big\lfloor \frac T2 \big\rfloor$ (the floor to cater for odd numbers) we have $$a_T=L_{0,T}a_0+\color{red}{\left(\sum_{r=0}^{\big\lfloor \frac T2 \big\rfloor}L_{r,T-r}s_{r+1}\right)}+W_T s_T$$
This is the best guess from the information provided.
(By the way, what do the variables represent?)