# Reference request for some determinants of binomial coefficients

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.

• 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. – Somos Oct 1 '18 at 21:37
• 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. – T. Amdeberhan Oct 2 '18 at 18:12

• 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. – Johann Cigler Oct 2 '18 at 9:20