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A203990 Symmetric matrix based on f(i,j) = (i+j)*min(i,j), by antidiagonals.

%I A203990 #17 Sep 08 2022 08:46:01
%S A203990 2,3,3,4,8,4,5,10,10,5,6,12,18,12,6,7,14,21,21,14,7,8,16,24,32,24,16,
%T A203990 8,9,18,27,36,36,27,18,9,10,20,30,40,50,40,30,20,10,11,22,33,44,55,55,
%U A203990 44,33,22,11,12,24,36,48,60,72,60,48,36,24,12,13,26,39,52,65,78,78,65,52,39,26,13
%N A203990 Symmetric matrix based on f(i,j) = (i+j)*min(i,j), by antidiagonals.
%C A203990 This sequence represents the matrix M given by f(i,j) = (i+j)*min{i,j} for i >= 1 and j >= 1.
%C A203990 See A203991 for characteristic polynomials of principal submatrices of M, with interlacing zeros.
%H A203990 G. C. Greubel, <a href="/A203990/b203990.txt">Table of n, a(n) for the first 100 rows, flattened</a>
%e A203990 Northwest corner:
%e A203990   2,  3,  4,  5,  6,  7
%e A203990   3,  8, 10, 12, 14, 16
%e A203990   4, 10, 18, 21, 24, 27
%e A203990   5, 12, 21, 32, 36, 40
%t A203990 (* First program *)
%t A203990 f[i_, j_] := (i + j) Min[i, j];
%t A203990 m[n_] := Table[f[i, j], {i, 1, n}, {j, 1, n}]
%t A203990 TableForm[m[6]] (* 6x6 principal submatrix *)
%t A203990 Flatten[Table[f[i, n + 1 - i], {n, 1, 12}, {i, 1, n}]]  (* A203990 *)
%t A203990 p[n_] := CharacteristicPolynomial[m[n], x];
%t A203990 c[n_] := CoefficientList[p[n], x]
%t A203990 TableForm[Flatten[Table[p[n], {n, 1, 10}]]]
%t A203990 Table[c[n], {n, 1, 12}]
%t A203990 Flatten[%]              (* A203991 *)
%t A203990 TableForm[Table[c[n], {n, 1, 10}]]
%t A203990 (* Second program *)
%t A203990 Table[(n+1)*Min[n-k+1, k], {n,15}, {k,n}]//Flatten (* _G. C. Greubel_, Jul 23 2019 *)
%o A203990 (PARI) for(n=1,15, for(k=1,n, print1((n+1)*min(n-k+1,k), ", "))) \\ _G. C. Greubel_, Jul 23 2019
%o A203990 (Magma) [(n+1)*Min(n-k+1,k): k in [1..n], n in [1..15]]; // _G. C. Greubel_, Jul 23 2019
%o A203990 (Sage) [[(n+1)*min(n-k+1,k) for n in (1..n)] for n in (1..15)] # _G. C. Greubel_, Jul 23 2019
%o A203990 (GAP) Flat(List([1..15], n-> List([1..n], k-> (n+1)*Minimum(n-k+1,k) ))); # _G. C. Greubel_, Jul 23 2019
%Y A203990 Cf. A203991, A202453.
%K A203990 nonn,tabl
%O A203990 1,1
%A A203990 _Clark Kimberling_, Jan 09 2012