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A368012 a(n) is the permanent of the n-th order Hankel matrix of Catalan numbers M(n) whose generic element is given by M(i,j) = A000108(i+j) with i,j = 0, ..., n-1.

%I A368012 #26 Dec 23 2023 12:37:59
%S A368012 1,1,3,95,38057,207372681,15977248385955,17828166968924572623,
%T A368012 292842668371666277607183121,71645110588632775032727941092738473,
%U A368012 263399284865064400938403105805219201386749363,14653009564320804036813733761485114583670416021283903839,12403293423772370760211339634714413308535752478944832963336911564521
%N A368012 a(n) is the permanent of the n-th order Hankel matrix of Catalan numbers M(n) whose generic element is given by M(i,j) = A000108(i+j) with i,j = 0, ..., n-1.
%H A368012 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 A368012 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 A368012 Wikipedia, <a href="https://en.wikipedia.org/wiki/Hankel_matrix">Hankel matrix</a>.
%F A368012 Det(M(n)) = 1 (see Mays and Wojciechowski, 2000).
%e A368012 a(4) = 38057:
%e A368012   1,  1,  2,   5;
%e A368012   1,  2,  5,  14;
%e A368012   2,  5, 14,  42;
%e A368012   5, 14, 42, 132.
%t A368012 Join[{1},Table[Permanent[Table[CatalanNumber[i+j],{i,0,n-1},{j,0,n-1}]],{n,12}]]
%o A368012 (PARI) C(n) = binomial(2*n, n)/(n+1); \\ A000108
%o A368012 a(n) = matpermanent(matrix(n, n, i, j, C(i+j-2))); \\ _Michel Marcus_, Dec 11 2023
%Y A368012 Cf. A000108.
%Y A368012 Cf. A278843, A368019, A368021, A368022, A368023, A368024, A368025.
%Y A368012 Column k=0 of A368026.
%K A368012 nonn
%O A368012 0,3
%A A368012 _Stefano Spezia_, Dec 08 2023