cp's OEIS Frontend

A296056 Determinant of the inverse of the matrix A_n, where A_n is the n X n matrix defined by A_n[i,j] = 1/C(i+j-2) for 1 <= i,j <= n, and C(k) is the k-th Catalan number (A000108).

%I A296056 #45 May 19 2020 03:11:55
%S A296056 1,-2,-1400,-679140000,-122489812645200000,
%T A296056 -6931927717187904217987200000,
%U A296056 -114287375178291587421201860354580633600000,-527655997339226839875614785993553970321322576128000000000,-666218073328701414704702576237379472614149140939534461737723520000000000000
%N A296056 Determinant of the inverse of the matrix A_n, where A_n is the n X n matrix defined by A_n[i,j] = 1/C(i+j-2) for 1 <= i,j <= n, and C(k) is the k-th Catalan number (A000108).
%C A296056 It is conjectured that a(n) is an integer for all n.
%C A296056 The contributor suggests the name "Catbert matrix" for the matrix A_n, based on its similarity to the Hilbert matrix and its relation to the Catalan numbers.
%H A296056 Tom Richardson, <a href="/A296056/b296056.txt">Table of n, a(n) for n = 1..29</a>
%H A296056 Tom Richardson, <a href="/A296056/a296056_2.txt">Table of n, a(n) for n = 1..100</a>
%H A296056 Thomas M. Richardson, <a href="https://arxiv.org/abs/2005.08939">Catalan Numbers and Jacobi Polynomials</a>, arXiv:2005.08939 [math.CO], 2020.
%F A296056 a(n) ~ -c * 16^(n*(n-1)) / (3^n * Pi^n * n^(27/8)), where c = 3*A^(3/2) / (2^(7/6) * exp(1/8) * sqrt(Pi)) = 0.9662886794923866798595701447717791386557874..., where A is the Glaisher-Kinkelin constant A074962. - _Vaclav Kotesovec_, May 19 2020
%t A296056 a[n_] := 1/Det@ Table[ 1/CatalanNumber[i + j -2], {i, n}, {j, n}]; Array[a, 9] (* _Robert G. Wilson v_, Jan 05 2018 *)
%t A296056 Table[Product[4^(2*k + 1) * (4*k - 1)/6 * Binomial[2*k - 3/2, k] * Binomial[2*k - 3/2, k + 1], {k, 0, n - 1}], {n, 1, 10}] (* _Vaclav Kotesovec_, May 19 2020 *)
%o A296056 (PARI) a(n) = 1/matdet(matrix(n,n,i,j,(i+j-1)/binomial(2*i+2*j-4,i+j-2)))
%Y A296056 Cf. A000108, A005249, A062381.
%K A296056 sign
%O A296056 1,2
%A A296056 _Tom Richardson_, Dec 03 2017