Generating function for binomial coefficients $\binom{2n+k}{n}$ with fixed $k$

Question

Prove that $$ \frac{1}{\sqrt{1-4t}} \left(\frac{1-\sqrt{1-4t}}{2t}\right)^k = \sum\limits_{n=0}^{\infty}\binom{2n+k}{n}t^n, \quad \forall k\in\mathbb{N}. $$ I tried already by induction over $k$ but i have problems showing the statement holds for $k=0$ or $k=1$.

Answer 1

The methodology is that you have got to make the brackets as the form expandable as the sum of polynomials with binomial coefficients And guess the clues through differentiation. It works no matter what the k is