This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A368025 #31 Nov 16 2024 19:15:01
%S A368025 1,1,1,1,1,1,1,1,2,1,1,1,3,5,1,1,1,4,14,14,1,1,1,5,30,84,42,1,1,1,6,
%T A368025 55,330,594,132,1,1,1,7,91,1001,4719,4719,429,1,1,1,8,140,2548,26026,
%U A368025 81796,40898,1430,1,1,1,9,204,5712,111384,884884,1643356,379236,4862,1
%N A368025 Array read by ascending antidiagonals: A(n,k) is the determinant of the n-th order Hankel matrix of Catalan numbers M(n) whose generic element is given by M(i,j) = A000108(i+j+k) with i,j = 0, ..., n-1.
%C A368025 This array is a variant of the triangles A078920 and A123352 extended to the trivial cases (here for k=0).
%H A368025 Arthur T. Benjamin, Naiomi T. Cameron, Jennifer J. Quinn, and Carl R. Yerger, <a href="https://combinatorialpress.com/cn/arch/vol200/">Catalan determinants-a combinatorial approach</a>, Congressus Numerantium 200, 27-34 (2010). <a href="https://www.researchgate.net/publication/249812385_Catalan_determinants-a_combinatorial_approach">On ResearchGate</a>.
%H A368025 Jishe Feng, <a href="https://arxiv.org/abs/2010.06586">The explicit formula of Hankel determinant with Catalan elements</a>, arXiv:2010.06586 [math.GM], 2020.
%H A368025 M. E. Mays and Jerzy Wojciechowski, <a href="https://doi.org/10.1016/S0012-365X(99)00140-5">A determinant property of Catalan numbers</a>. Discrete Math. 211, No. 1-3, 125-133 (2000).
%H A368025 Wikipedia, <a href="https://en.wikipedia.org/wiki/Hankel_matrix">Hankel matrix</a>.
%F A368025 For an explicit formula of A(n,k), see equation (5) in Feng, 2020.
%F A368025 A(n,2) = n + 1.
%F A368025 A(n,3) = A000330(n+1).
%F A368025 A(n,4) = A006858(n+1).
%F A368025 A(n,5) = A091962(n+1).
%F A368025 Diagonal: A(n,n) = A123352(2*n-1,n-1) = A355400(n).
%e A368025 The array begins:
%e A368025 1, 1, 1, 1, 1, 1, 1, ...
%e A368025 1, 1, 2, 5, 14, 42, 132, ...
%e A368025 1, 1, 3, 14, 84, 594, 4719, ...
%e A368025 1, 1, 4, 30, 330, 4719, 81796, ...
%e A368025 1, 1, 5, 55, 1001, 26026, 884884, ...
%e A368025 1, 1, 6, 91, 2548, 111384, 6852768, ...
%e A368025 1, 1, 7, 140, 5712, 395352, 41314284, ...
%e A368025 ...
%p A368025 A:= proc(n, k) option remember; `if`(k=0, 1, 2^n*mul(
%p A368025 (2*(k-i)+2*n-3)/(k+2*n-1-i), i=0..n-1)*A(n, k-1))
%p A368025 end:
%p A368025 seq(seq(A(d-k, k), k=0..d), d=0..10); # _Alois P. Heinz_, Dec 20 2023
%t A368025 A[n_,k_]:=If[n==0,1,Det[Table[CatalanNumber[i+j+k],{i,0,n-1},{j,0,n-1}]]]; Table[A[n-k,k],{n,0,11},{k,0,n}]//Flatten
%Y A368025 Cf. A000108 (n=1), A005700 (n=2), A006149 (n=3), A006150 (n=4), A006151 (n=5).
%Y A368025 Cf. A000012 (k=0 or k=1 or n=0), A000330, A078920, A091962, A123352, A335857 (k=6).
%Y A368025 Cf. A355400, A368026 (permanent), A378112.
%Y A368025 Antidiagonal sums give A355503.
%K A368025 nonn,tabl
%O A368025 0,9
%A A368025 _Stefano Spezia_, Dec 08 2023