Expanding the generating function of the Fibonacci numbers to find a cute formula

Question

$F_0=1$, $F_1=1$, $F_n=F_{n-1}+F_{n-2}$. The generating function is $-\frac{1}{x^2+x-1}$. I have to expand it to prove that $F_n=\sum_k\binom{k}{n-k}$. Could you help me please?

Answer 1

Expanding as a geometric series and then using the Binomial Theorem, we obtain $$\sum_{k=0}^\infty (x^2+x)^k=\sum_{k=0}^\infty \sum_{l=0}^k{k\choose l}x^{2l+(k-l)}=\sum_{n=0}^\infty\left(\sum_{k+l=n}{k\choose l}\right)x^n$$ Rewrite the inner sum's index $l=n-k$ so that equating coefficients with $\sum F_n x^n$ gives $$F_n=\sum_{k} {k\choose n-k}.$$