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A203993 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of {|i-j}, (A049581).

%I A203993 #20 Feb 16 2025 08:33:16
%S A203993 0,-1,-1,0,1,4,6,0,-1,-12,-32,-20,0,1,32,120,140,50,0,-1,-80,-384,
%T A203993 -648,-448,-105,0,1,192,1120,2464,2520,1176,196,0,-1,-448,-3072,-8320,
%U A203993 -11264,-7920,-2688,-336,0,1,1024,8064,25920,43680
%N A203993 Array:  row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of {|i-j}, (A049581).
%C A203993 Let p(n)=p(n,x) be the characteristic polynomial of the n-th principal submatrix.  The zeros of p(n) are real, and they interlace the zeros of p(n+1).  See A202605 for a guide to related sequences.
%C A203993 Also the coefficients of the detour and distance polynomials of the n-path graph P_n. - _Eric W. Weisstein_, Apr 07 2017
%C A203993 p(n,x) = (-x)^n*(x*(1 + T(n, 1+1/x)) - n*S(n-1, 2*(1+1/x)))/(2*x), with the Chebyshev polynomials S (A049310) and T (A053120). This is the rewritten formula given below in the Mathematica program by Weisstein. - _Wolfdieter Lang_, Feb 02 2018
%D A203993 (For references regarding interlacing roots, see A202605.)
%H A203993 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DetourPolynomial.html">Detour Polynomial</a>
%H A203993 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DistancePolynomial.html">Distance Polynomial</a>
%H A203993 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PathGraph.html">Path Graph</a>
%H A203993 <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%F A203993 T(n, k) = [x^k] p(n,x), with p(n,x) = Determinant(M_n - x*1_n), with the n x n matrix M_n with entries M_n(i, j) = |i-j|, for n >= 1, k = 0, 1, ..., n. For p(n,x) see a comment above and the Mathematica formulas by Weisstein.- _Wolfdieter Lang_, Feb 02 2018
%e A203993 The array T (a table if row n=0 is by convention put to 0) begins:
%e A203993 n\k     0      1      2       3       4       5      6      7     8    9  10 ...
%e A203993 (0:     0)
%e A203993 1:      0     -1
%e A203993 2:     -1      0      1
%e A203993 3:      4      6      0      -1
%e A203993 4:    -12    -32    -20       0       1
%e A203993 5:     32    120    140      50       0      -1
%e A203993 6:    -80   -384   -648    -448    -105       0      1
%e A203993 7:    192   1120   2464    2520    1176     196      0     -1
%e A203993 8:   -448  -3072  -8320  -11264   -7920   -2688   -336      0     1
%e A203993 9:   1024   8064  25920   43680   41184   21384   5544    540     0   -1
%e A203993 10: -2304 -20480 -76160 -153600 -182000 -128128 -51480 -10560  -825    0   1
%e A203993 ... reformatted and extended. - _Wolfdieter Lang_, Feb 02 2018
%t A203993 (* begin*)
%t A203993 f[i_, j_] := Abs[i - j];
%t A203993 m[n_] := Table[f[i, j], {i, 1, n}, {j, 1, n}]
%t A203993 TableForm[m[6]] (* 6x6 principal submatrix *)
%t A203993 Flatten[Table[f[i, n + 1 - i],
%t A203993 {n, 1, 12}, {i, 1, n}]]  (* A049581 *)
%t A203993 p[n_] := CharacteristicPolynomial[m[n], x];
%t A203993 c[n_] := CoefficientList[p[n], x]
%t A203993 TableForm[Flatten[Table[p[n], {n, 1, 10}]]]
%t A203993 Table[c[n], {n, 1, 12}]
%t A203993 Flatten[%]    (* A203993 *)
%t A203993 TableForm[Table[c[n], {n, 1, 10}]]
%t A203993 (* end *)
%t A203993 CoefficientList[Table[CharacteristicPolynomial[SparseArray[{i_, j_} :> Abs[i - j], n], x], {n, 10}], x] //Flatten (* _Eric W. Weisstein_, Apr 07 2017 *)
%t A203993 CoefficientList[Table[((-x)^n (x + x ChebyshevT[2 n, Sqrt[1 + 1/(2 x)]] - n ChebyshevU[n - 1, 1 + 1/x]))/(2 x), {n, 10}], x] // Flatten (* _Eric W. Weisstein_, Apr 07 2017 *)
%t A203993 CoefficientList[Table[1/4 (2 (-x)^n + (-1 - x - Sqrt[1 + 2 x])^n + (-1 - x + Sqrt[1 + 2 x])^n + (n (-(-1 - x - Sqrt[1 + 2 x])^n + (-1 - x + Sqrt[1 + 2 x])^n))/Sqrt[1 + 2 x]), {n, 10}], x] // Flatten (* _Eric W. Weisstein_, Apr 07 2017 *)
%t A203993 CoefficientList[LinearRecurrence[{-4 - 5 x, -2 (2 + 6 x + 5 x^2), -2 x (2 + 6 x + 5 x^2), -x^3 (4 + 5 x), -x^5}, {-x, (-1 + x) (1 + x), -(2 + x) (-2 - 2 x + x^2), (-6 - 4 x + x^2) (2 + 4 x + x^2), -(4 + 6 x + x^2) (-8 - 18 x - 6 x^2 + x^3)}, 10], x] // Flatten (* _Eric W. Weisstein_, Apr 07 2017 *)
%Y A203993 Cf. A049310, A049581, A053120, A085750 (column k=0, Det(M_n)), A166445(n-1) (alternating row sums), A202605.
%K A203993 tabl,sign,easy
%O A203993 1,6
%A A203993 _Clark Kimberling_, Jan 09 2012