The paper proves Knuth's old sum in two lines and derives a v-parameter generalization with many corollary identities.
Miki\'c, A proof of a famous identity concerning the convolution of the central binomial coefficients, Journal of Integer Sequences 19 (2016), 1--10, Article 16.6.6
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A Short Proof of Knuth's Old Sum
The paper proves Knuth's old sum in two lines and derives a v-parameter generalization with many corollary identities.