\displaystyle f(x)=\sum_{n\geq 0}\sum_{k=0}^{n}{n-k+1\choose k}x^{n}\\
=\sum_{k\geq 0}\sum_{n\geq k}{n-k+1\choose k}x^{n}\\
=\sum_{k\geq 0}\sum_{n\geq 0}{n+1\choose k}x^{n+k}\\
=\sum_{n\geq 0}x^{n}\sum_{k\geq 0}{n+1\choose k}x^{k}\\
=\sum_{n\geq 0}x^{n}(1+x)^{n+1}\\
=(1+x)\sum_{n\geq 0}(x+x^{2})^{n}\\
=\frac{1+x}{1-x-x^{2}}