The harmonic numbers are given by $$H_n=\sum_{k=1}^n\frac{1}{k}.$$ Numerical calculation suggests $$ \sum_{k=1}^{n}(-1)^k{n\choose k}{n+k\choose k}\sum_{i=1}^{k}\frac{1}{n+i}=(-1)^nH_n. $$ I can not give a proof and want to know if this identity exists in the literature. If not, how can we prove it?

Hints, references or proof are all welcome.