I asked [this question on math.SE][1] before, but did get not get an answer. Therefore I hope it is ok to post it here on this site..

Let $q \in \mathbb{R}_+$ and $0 < w_1 \leq w_2 \leq \ldots w_d \in \mathbb{R}_+$. Define
$$A_w(d,q):=\left\{{\bf k} \in \mathbb{N}_0^d: \sum_{j=1}^d w_j k_j \leq q\right\}$$
denote the number of non-negative integer points in the $\ell_1$-ellipse with semi-axes of length $\frac{q}{w_j}$ (which is a simplex).
Similarly, 
$$B_w(d,q):=\left\{{\bf k} \in \mathbb{N}_+^d: \sum_{j=1}^d w_j k_j \leq q\right\}$$
denotes the number of positive integer points in this simplex.

My question is: Is there a relationship between $A_w(d,q)$ and $B_w(d,q)$ in the sense that if one could obtain an exact formula for $A_w(d,q)$ there also would be an exact formula for $B_w(d,q)$? For example something like
$$B_w(d,q) = A_\tilde{w}(d,\tilde{q}) .$$

Thanks!

[1]: https://math.stackexchange.com/questions/1405826/integer-points-in-simplex