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Array read by ascending antidiagonals: A(n, k) 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+k) with i,j = 0, ..., n-1.
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%I #18 Dec 23 2023 12:55:49

%S 1,1,1,3,1,1,95,9,2,1,38057,979,53,5,1,207372681,1417675,19148,406,14,

%T 1,15977248385955,28665184527,97432285,490614,3612,42,1,

%U 17828166968924572623,8325587326635565,7146659536022,8755482505,14798454,35442,132,1,292842668371666277607183121,35389363346700690999467,7683122105385590481,2318987094804471,930744290905,499114473,372801,429,1

%N Array read by ascending antidiagonals: A(n, k) 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+k) with i,j = 0, ..., n-1.

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

%e The array begins:

%e 1, 1, 1, 1, 1, ...

%e 1, 1, 2, 5, 14, ...

%e 3, 9, 53, 406, 3612, ...

%e 95, 979, 19148, 490614, 14798454, ...

%e 38057, 1417675, 97432285, 8755482505, 930744290905, ...

%e ...

%p with(LinearAlgebra):

%p C:= proc(n) option remember; binomial(2*n, n)/(n+1) end:

%p A:= (n, k)-> `if`(n=0, 1, Permanent(Matrix(n, (i, j)-> C(i+j+k-2)))):

%p seq(seq(A(d-k, k), k=0..d), d=0..8); # _Alois P. Heinz_, Dec 20 2023

%t A[n_,k_]:=If[n==0,1,Permanent[Table[CatalanNumber[i+j+k],{i,0,n-1},{j,0,n-1}]]]; Table[A[n-k,k],{n,0,8},{k,0,n}]//Flatten

%Y Cf. A000012 (n=0), A000108 (n=1).

%Y Cf. A368012 (k=0), A368019 (k=1), A278843 (k=2), A368021 (k=3), A368022 (k=4), A368023 (k=5), A368024 (k=6).

%Y Cf. A368025 (determinant), A368298 (diagonal).

%K nonn,tabl

%O 0,4

%A _Stefano Spezia_, Dec 08 2023