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.

  • 2
    $\begingroup$ 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. $\endgroup$ Oct 2 '18 at 18:12

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.

  • 1
    $\begingroup$ Thank you. I know your paper and I have in fact used the Condensation Method to compute the more general determinant $\det\left(\binom{i+j+x+k-2}{i-j+k}\right)_{0 \leq i,j\leq {n-1}}.$ 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. $\endgroup$ Oct 2 '18 at 9:20

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.