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 A203990 Symmetric matrix based on f(i,j) = (i+j)*min(i,j), by antidiagonals. 4

%I #17 Sep 08 2022 08:46:01

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

%C This sequence represents the matrix M given by f(i,j) = (i+j)*min{i,j} for i >= 1 and j >= 1.

%C See A203991 for characteristic polynomials of principal submatrices of M, with interlacing zeros.

%H G. C. Greubel, <a href="/A203990/b203990.txt">Table of n, a(n) for the first 100 rows, flattened</a>

%e Northwest corner:

%e 2, 3, 4, 5, 6, 7

%e 3, 8, 10, 12, 14, 16

%e 4, 10, 18, 21, 24, 27

%e 5, 12, 21, 32, 36, 40

%t (* First program *)

%t f[i_, j_] := (i + j) Min[i, j];

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

%t TableForm[m[6]] (* 6x6 principal submatrix *)

%t Flatten[Table[f[i, n + 1 - i], {n, 1, 12}, {i, 1, n}]] (* A203990 *)

%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[%] (* A203991 *)

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

%t (* Second program *)

%t Table[(n+1)*Min[n-k+1, k], {n,15}, {k,n}]//Flatten (* _G. C. Greubel_, Jul 23 2019 *)

%o (PARI) for(n=1,15, for(k=1,n, print1((n+1)*min(n-k+1,k), ", "))) \\ _G. C. Greubel_, Jul 23 2019

%o (Magma) [(n+1)*Min(n-k+1,k): k in [1..n], n in [1..15]]; // _G. C. Greubel_, Jul 23 2019

%o (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 (GAP) Flat(List([1..15], n-> List([1..n], k-> (n+1)*Minimum(n-k+1,k) ))); # _G. C. Greubel_, Jul 23 2019

%Y Cf. A203991, A202453.

%K nonn,tabl

%O 1,1

%A _Clark Kimberling_, Jan 09 2012

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