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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+4) with i,j = 0, ..., n-1.
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%I #16 Dec 23 2023 12:38:15

%S 1,14,3612,14798454,930744290905,891107801867703108,

%T 12977575456694246217097712,2880177942851157900010279504962852,

%U 9767068920318918290905853040035029840419305,507521146153330160633968276251306280235282377512091202,405202219609475677155580649938116235991326716758748940659564085180

%N 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+4) with i,j = 0, ..., n-1.

%H 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 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 Wikipedia, <a href="https://en.wikipedia.org/wiki/Hankel_matrix">Hankel matrix</a>.

%F Det(M(n)) = A006858(n+1).

%e a(4) = 930744290905:

%e 14, 42, 132, 429;

%e 42, 132, 429, 1430;

%e 132, 429, 1430, 4862;

%e 429, 1430, 4862, 16796.

%t Join[{1},Table[Permanent[Table[CatalanNumber[i+j+4],{i,0,n-1},{j,0,n-1}]],{n,10}]]

%o (PARI) C(n) = binomial(2*n, n)/(n+1); \\ A000108

%o a(n) = matpermanent(matrix(n, n, i, j, C(i+j+2))); \\ _Michel Marcus_, Dec 11 2023

%Y Cf. A000108, A006858.

%Y Cf. A278843, A368012, A368019, A368021, A368023, A368024, A368025.

%Y Column k=4 of A368026.

%K nonn

%O 0,2

%A _Stefano Spezia_, Dec 08 2023