Let $C_{n}=\frac{1}{n+1}\binom{2n}{n}$ be a Catalan number and $n$ and $k$ be non-negative integers. Consider the following identities: $$ \det\left(\binom{i+j+k}{2j}\right)_{0 \leq i,j\leq {n-1}}=\det\left(C_{n+i+j}\right)_{0 \leq i,j\leq {k-1}}$$ and $$ \det\left(\binom{i+j+k+1}{2j+1}\right)_{0 \leq i,j\leq {n-1}}=\det\left(C_{n+1+i+j}\right)_{0 \leq i,j\leq {k-1}}.$$ The determinants on the left-hand side can be computed using Theorem 27 of “Advanced Determinant Calculus” by C. Krattenthaler and the determinants on the right-hand side are well known. Due to their special form it is very probable that (some of) these determinants (with or without the identification with the right-hand side) have been (re)discovered several times.

I am very interested to get references to books or papers where these special determinants (at least for the cases $k=1$ or $k=2$ have been studied.



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