The blog is about the ordinary generating function(OGF), and it is updating.

The ordinary generating function of a sequence ($A_n$) is the formal power series:
$$
A(z)=\sum_{n=0}^{\infty}A_nz^n
$$
The ordinary generating function of a combinatorial class $\cal{A}$ is the generating function of numbers $A_n=\rm{card}(\cal{A_n})$.Equivalently, the OGF can also be like:
$$
A(z)=\sum_{\alpha\in\cal{A}}z^{|(\alpha)|}
$$
Here is some useful formulas:

1.1 Sum
$$
\cal{A}=\cal{B}+\cal{C}\Longrightarrow \it{A}(z)=B(z)+C(z)\
$$

Proof. It follows directly from the combinatorial form of OGF:

$A(z)=\sum_{\alpha\in\cal{A}}z^{|(\alpha)|}=\sum_{\alpha\in\cal{B}}z^{|(\alpha)|}+\sum_{\alpha\in\cal{C}}z^{|(\alpha)|}=B(z)+C(z)$

1.2 Cartesian product
$$
\cal{A}=\cal{B}\times\cal{C}\Longrightarrow \it{A}(z)=B(z)C(z)\
$$
Proof.
$$
A(z)=\sum_{\alpha\in \cal{A}}z^{|\alpha|}=\sum_{(\beta,\gamma)\in(\cal{B}\times\cal{C})}z^{|\beta|+|\gamma|}=(\sum_{\beta\in\cal{B}}z^{|\beta|})(\sum_{\gamma\in\cal{C}}z^{|\gamma|})=B(z)C(z)
$$
By considering all possibilities, it is related by the convolution relation:
$$
A_n=\sum_{k=0}^{n}B_kC_{n-k}
$$