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Let $C_{n}=\binom{2n}{n}\frac{1}{n+1}$ be a Catalan number. I am interested in books or papers where the following identities occur: $$\det\left(\binom{i+j+1}{i-j+1}\right)_{0 \leq i,j\leq {n-1}}=C_{n}$$ and $$\det\left(\binom{i+j+2}{i-j+1}\right)_{0 \leq i,j\leq {n-1}}=C_{n+1}$$ I have already posted a more general question in https://math.stackexchange.com/questions/2928770/reference-request-for-some-determinants-of-binomial-coefficients, but did not get an answer.

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    $\begingroup$ Let $\, a_{n,k} := \det (A_{n,k}) \,$ where $\, A_{n,k} := \{{i+j+k-1 \choose i-j+1}\}_{1\le i,j\le n}.\,$ Then $\, a_{n,k} = (k+1) { 2n+k \choose n }/(n+k+1). \,$ See OEIS sequence A054445. $\endgroup$ – Somos Oct 1 '18 at 21:37
  • $\begingroup$ Just for the record and (perhaps) some insight: $\det\left(\binom{i+j+k}{i-j+1}\right)=C_n*C_n*\cdots *C_n$, a (discrete) convolution of $k$ Catalans. $\endgroup$ – T. Amdeberhan Oct 2 '18 at 18:12
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Here is a paper https://www.sciencedirect.com/science/article/pii/S0196885801907328 Advances in Applied Mathematics Volume 27, Issues 2–3, August 2001, Pages 225-230.

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    $\begingroup$ Thank you. I know your paper and I have in fact used the Condensation Method to compute the more general determinant $\det\left(\binom{i+j+x+k-2}{i-j+k}\right)_{0 \leq i,j\leq {n-1}}.$ But the two special cases are so simple that I suppose that these must be old results and I want to know who has found them first. $\endgroup$ – Johann Cigler Oct 2 '18 at 9:20

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