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For fixed $r$ you may expand ${2k-n+a\choose r}$ as a polynomial in $k$ (whose coefficients are polynomials in $n-a$) as ${2k-n+a\choose r}=\sum_{i=0}^r c_i {k\choose i}$, substitute this expansion to your sum, use $${n\choose k}{k\choose i} ={n\choose i} {n-i\choose k-i},$$ and finally apply Vandermonde convolution $$\sum_{k=i}^n {n-i\choose k-i}{n\choose k+a}={2n-i\choose n+a}$$ to express your sum as $$\sum_{i=0}^rc_i{2n-i\choose n+a}.$$