Basically I'd like to know how to derive the generation function for the finite Coxeter group of type $D_k$ to be familiar with notes in OEIS A162288:
According to formula section:
'The growth series for the finite Coxeter group of type $D_k$ ($k\ge3$) has $G.f. = \prod_{i} (1-x^{m_i})/(1-x)$ where the $m_i$ are $[1,3,5,...,2k-3,k-1]$.
On the other hand Mathematica section of the sequence has a different formula: $(1-x^n) \prod_{k=1}^{n-1} (1-x^{2k})/(1-x)^n$.
In addition to that some draft correction has been provided recently: 'The growth series for the finite Coxeter group of type $D_k$ ($k \ge3$) has $g.f. = \prod_i ((1-x^{m_i})/(1-x))$ where the $m_i$ are $[2, 4, 6, ..., 2k-2; k]$.
Clarification on that topic to highlight the right formulas are highly welcomed. Derivation of that formulas (from one to another) is also of right help.