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A368019 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+1) with i,j = 0, ..., n-1.

%I A368019 #16 Dec 23 2023 12:37:50
%S A368019 1,1,9,979,1417675,28665184527,8325587326635565,
%T A368019 35389363346700690999467,2230867495754739989535874468003,
%U A368019 2106171270085074740753132799048111935155,30007898337707083458776293190436074888346472515407,6491219550166075876771081259839537013093735814742318424677245
%N A368019 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+1) with i,j = 0, ..., n-1.
%H A368019 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 A368019 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 A368019 Wikipedia, <a href="https://en.wikipedia.org/wiki/Hankel_matrix">Hankel matrix</a>.
%F A368019 Det(M(n)) = 1 (see Mays and Wojciechowski, 2000).
%e A368019 a(4) = 1417675:
%e A368019    1,  2,   5,  14;
%e A368019    2,  5,  14,  42;
%e A368019    5, 14,  42, 132;
%e A368019   14, 42, 132, 429.
%t A368019 Join[{1},Table[Permanent[Table[CatalanNumber[i+j+1],{i,0,n-1},{j,0,n-1}]],{n,11}]]
%o A368019 (PARI) C(n) = binomial(2*n,n)/(n+1); \\ A000108
%o A368019 a(n) = matpermanent(matrix(n,n,i,j,C(i+j-1))); \\ _Michel Marcus_, Dec 09 2023
%Y A368019 Cf. A000108.
%Y A368019 Cf. A278843, A368012, A368021, A368022, A368023, A368024, A368025.
%Y A368019 Column k=1 of A368026.
%K A368019 nonn
%O A368019 0,3
%A A368019 _Stefano Spezia_, Dec 08 2023