Suppose $H(x)$ is a generating function of a sequence $(h_n)$. How do I go about expressing following sequences in terms of $H_n$?
$$a_n = {n}\cdot{h_n}$$ $$b_n = \sum_{k=0}^n(h_n)$$ $$c_n = \left\{\begin{matrix} h_n, & n \text{ is even}\\ 0, & n \text{ is odd} \end{matrix}\right.$$ $$d_n = n^k\cdot{h_n},\;\;(\text{for some constant }k\in\mathbb{N})$$
I have found different materials online that help me solve those cases, but I am having some trouble connecting these relations into some intuition, understanding an intuitive connection between a generating function and the sequence, and how some alteration of the sequence affects generating function and vise-versa.
Suppose that $$ H(x)=\sum_{n=0}^\infty h_n x^n. $$ Then $$ x\frac{dH}{dx}=\sum_{n=0}^\infty nh_nx^n; \quad [(xD)^kH](x)=\sum_{n=0}^\infty n^k h_nx^n $$ where $[(xD)f](x)=x\frac{df}{dx}$. Also $$ \frac{H(x)}{1-x}=\sum_{n=0}^\infty\left(\sum_{k=0}^nh_{k}\right) x^n. $$ Finally $$ \sum_{n-0}^\infty c_nx^n=\frac{H(x)+H(-x)}{2}. $$