Is there any function in python or sage to obtain explicit formula of coefficients for generating functions. For example, Catalan generating function is given as
$${\displaystyle c(x)={\dfrac {1-{\sqrt {1-4x}}}{2x}}}$$ And I need to obtain general formula for $n$th coefficient:
$${\displaystyle C_{n}={\dfrac {1}{2n+1}}{2n+1 \choose n}}$$
In this case you want the generalized binomial theorem which is
$(1+x)^a =\sum_{n=0}^{\infty} \binom{a}{n}x^n $ where $\binom{a}{n} =\dfrac{\prod_{k=0}^{n-1}(a-k)}{n!} $.
For your case $a=\dfrac12$ so, for $n \ge 1$,
$\begin{array}\\ \binom{\frac12}{n} &=\dfrac{\prod_{k=0}^{n-1}(\frac12-k)}{n!}\\ &=\dfrac{\prod_{k=0}^{n-1}(\frac12(1-2k))}{n!}\\ &=\dfrac{(-1)^n\prod_{k=0}^{n-1}(2k-1)}{2^nn!}\\ &=\dfrac{(-1)^{n+1}\prod_{k=1}^{n-1}(2k-1)}{2^nn!}\\ &=\dfrac{(-1)^{n+1}\prod_{k=1}^{n-1}(2k-1)\prod_{k=1}^{n-1}(2k)}{2^nn!\prod_{k=1}^{n-1}(2k)}\\ &=\dfrac{(-1)^{n+1}\prod_{k=1}^{2n-2}k}{2^nn!2^{n-1}(n-1)!}\\ &=\dfrac{(-1)^{n+1}(2n-2)!}{2^{2n-1}n!(n-1)!}\\ \end{array} $
so
$\begin{array}\\ \sqrt {1-4x} &=\sum_{n=0}^{\infty}\binom{\frac12}{n}(-4x)^n\\ &=1+\sum_{n=1}^{\infty}\dfrac{(-1)^{n+1}(2n-2)!}{2^{2n-1}n!(n-1)!}(-1)^n4^nx^n\\ &=1-\sum_{n=1}^{\infty}\dfrac{2(2n-2)!}{n!(n-1)!}x^n\\ \end{array} $
so
$\begin{array}\\ \dfrac {1-\sqrt {1-4x}}{2x} &=\dfrac{2\sum_{n=1}^{\infty}\dfrac{(2n-2)!}{n!(n-1)!}x^n}{2x}\\ &=\sum_{n=1}^{\infty}\dfrac{(2n-2)!}{n!(n-1)!}x^{n-1}\\ &=\sum_{n=0}^{\infty}\dfrac{(2n)!}{n!(n+1)!}x^{n}\\ &=\sum_{n=0}^{\infty}\dfrac{(2n+1)!}{(2n+1)n!(n+1)!}x^{n}\\ &=\sum_{n=0}^{\infty}\dfrac1{2n+1}\binom{2n+1}{n}x^{n}\\ \end{array} $
I'll be glad to explain any of this you do not understand since it uses a number of standard manipulations you should get familiar with.