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Symmetric matrix based on f(i,j)=(i if i=j and 1 otherwise), by antidiagonals.
4

%I #5 Mar 30 2012 18:58:07

%S 1,1,1,1,2,1,1,1,1,1,1,1,3,1,1,1,1,1,1,1,1,1,1,1,4,1,1,1,1,1,1,1,1,1,

%T 1,1,1,1,1,1,5,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,6,1,1,1,1,1,1,1,

%U 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,7,1,1,1,1,1,1,1,1,1,1,1,1,1,1

%N Symmetric matrix based on f(i,j)=(i if i=j and 1 otherwise), by antidiagonals.

%C A204125 represents the matrix M given by f(i,j)=max([i/j],[j/i]) for i>=1 and j>=1. See A204126 for characteristic polynomials of principal submatrices of M, with interlacing zeros. See A204016 for a guide to other choices of M.

%e Northwest corner:

%e 1 1 1 1 1 1

%e 1 2 1 1 1 1

%e 1 1 3 1 1 1

%e 1 1 1 4 1 1

%e 1 1 1 1 5 1

%e 1 1 1 1 1 6

%t f[i_, j_] := 1; f[i_, i_] := i;

%t m[n_] := Table[f[i, j], {i, 1, n}, {j, 1, n}]

%t TableForm[m[8]] (* 8x8 principal submatrix *)

%t Flatten[Table[f[i, n + 1 - i],

%t {n, 1, 15}, {i, 1, n}]] (* A204125 *)

%t p[n_] := CharacteristicPolynomial[m[n], x];

%t c[n_] := CoefficientList[p[n], x]

%t TableForm[Flatten[Table[p[n], {n, 1, 10}]]]

%t Table[c[n], {n, 1, 12}]

%t Flatten[%] (* A204126 *)

%t TableForm[Table[c[n], {n, 1, 10}]]

%Y Cf. A204126, A204016, A202453.

%K nonn,tabl

%O 1,5

%A _Clark Kimberling_, Jan 11 2012