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A203994 Symmetric matrix based on f(i,j) = (i+j)*min{i,j}, by antidiagonals.

%I A203994 #13 Sep 08 2022 08:46:01
%S A203994 1,0,0,-1,1,-1,-2,0,0,-2,-3,-1,1,-1,-3,-4,-2,0,0,-2,-4,-5,-3,-1,1,-1,
%T A203994 -3,-5,-6,-4,-2,0,0,-2,-4,-6,-7,-5,-3,-1,1,-1,-3,-5,-7,-8,-6,-4,-2,0,
%U A203994 0,-2,-4,-6,-8,-9,-7,-5,-3,-1,1,-1,-3,-5,-7,-9
%N A203994 Symmetric matrix based on f(i,j) = (i+j)*min{i,j}, by antidiagonals.
%C A203994 A203994 represents the matrix M given by f(i,j) = min(i-j+1,j-i+1) for i >= 1 and j >= 1.  See A203995 for characteristic polynomials of principal submatrices of M, with interlacing zeros.
%H A203994 G. C. Greubel, <a href="/A203994/b203994.txt">Antidiagonal rows n = 1..100, flattened</a>
%e A203994 Northwest corner:
%e A203994    1    0   -1   -2   -3
%e A203994    0    1    0   -1   -2
%e A203994   -1    0    1    0   -1
%e A203994    2   -1    0    1    0
%t A203994 (* First program *)
%t A203994 f[i_, j_] := Min[i - j + 1, j - i + 1];
%t A203994 m[n_] := Table[f[i, j], {i, 1, n}, {j, 1, n}]
%t A203994 TableForm[m[6]] (* 6 X 6 principal submatrix *)
%t A203994 Flatten[Table[f[i, n + 1 - i],
%t A203994 {n, 1, 12}, {i, 1, n}]]  (* A203994 *)
%t A203994 p[n_] := CharacteristicPolynomial[m[n], x];
%t A203994 c[n_] := CoefficientList[p[n], x]
%t A203994 TableForm[Flatten[Table[p[n], {n, 1, 10}]]]
%t A203994 Table[c[n], {n, 1, 12}]
%t A203994 Flatten[%]    (* A203995 *)
%t A203994 TableForm[Table[c[n], {n, 1, 10}]]
%t A203994 (* Second program *)
%t A203994 Table[Min[2*k-n, n-2*k+2], {n,15}, {k,n}]//Flatten (* _G. C. Greubel_, Jul 23 2019 *)
%o A203994 (PARI) for(n=1,15, for(k=1,n, print1(min(2*k-n, n-2*k+2), ", "))) \\ _G. C. Greubel_, Jul 23 2019
%o A203994 (Magma) [Min(2*k-n, n-2*k+2): k in [1..n], n in [1..15]]; // _G. C. Greubel_, Jul 23 2019
%o A203994 (Sage) [[min(2*k-n, n-2*k+2) for k in (1..n)] for n in (1..15)] # _G. C. Greubel_, Jul 23 2019
%o A203994 (GAP) Flat(List([1..15], n-> List([1..n], k-> Minimum(2*k-n, n-2*k+2) ))); # _G. C. Greubel_, Jul 23 2019
%Y A203994 Cf. A203995, A202453.
%K A203994 tabl,sign
%O A203994 1,7
%A A203994 _Clark Kimberling_, Jan 09 2012