$[z^n]$ notation

Question

I have the following formula:

$[z^n]\frac{2+3z^2}{\sqrt{1-5z}}$

I'm not sure about the meaning of $[z^n]$. I don't have a definition for this in lecture notes, could someone clarify what exactly does it mean?

I used to use this notation for equivalence class, but it does not make sense to me in this context.

Answer 1

The expression

$$ [z^n]\frac{2+3z^2}{\sqrt{1-5z}} $$

means "The coefficient of $z^n$ in the Maclaurin series for $\frac{2+3z^2}{\sqrt{1-5z}}$."

For example,

$$ \frac{2+3z^2}{\sqrt{1-5z}} = 2 + 5z + \frac{87}{4}z^2 + \frac{685}{8}z^3 + \cdots, $$

so

$$ [z^3]\frac{2+3z^2}{\sqrt{1-5z}} = \frac{685}{8}. $$