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Is there a nice bijective proof of the fact that the determinant of the $(n+1)$-by-$(n+1)$ Hankel matrix whose respective entries are the central binomial coefficients $0 \choose 0$, $2 \choose 1$, $\...

Identity \begin{equation} \sum_{k=0}^n\binom nk^2 \binom k{n-m}=\binom nm \binom{n+m}m \tag{1} \end{equation} was used in an answer here. As shown in that answer, (1) easily reduces to \begin{...

In my paper Supercongruences involving dual sequences [Finite Fields Appl. 46(2017), 179-216], I gave a new symmetric identity which states that if $x+y=-1$ then $$\sum_{k=0}^n(-1)^k\binom xk^2\binom{...