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: $$\det\left(\binom{i+j+1}{ij+1}\right)_{0 \leq i,j\leq {n1}}=C_{n}$$ and $$\det\left(\binom{i+j+2}{ij+1}\right)_{0 \leq i,j\leq {n1}}=C_{n+1}$$ I have already posted a more general question in https://math.stackexchange.com/questions/2928770/referencerequestforsomedeterminantsofbinomialcoefficients, but did not get an answer.
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2$\begingroup$ Let $\, a_{n,k} := \det (A_{n,k}) \,$ where $\, A_{n,k} := \{{i+j+k1 \choose ij+1}\}_{1\le i,j\le n}.\,$ 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: $\det\left(\binom{i+j+k}{ij+1}\right)=C_n*C_n*\cdots *C_n$, 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 225230.

1$\begingroup$ Thank you. I know your paper and I have in fact used the Condensation Method to compute the more general determinant $\det\left(\binom{i+j+x+k2}{ij+k}\right)_{0 \leq i,j\leq {n1}}.$ 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