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

0, -1, -1, 0, 1, 4, 6, 0, -1, -12, -32, -20, 0, 1, 32, 120, 140, 50, 0, -1, -80, -384, -648, -448, -105, 0, 1, 192, 1120, 2464, 2520, 1176, 196, 0, -1, -448, -3072, -8320, -11264, -7920, -2688, -336, 0, 1, 1024, 8064, 25920, 43680

Offset

1,6

Comments

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.

Also the coefficients of the detour and distance polynomials of the n-path graph P_n. - Eric W. Weisstein, Apr 07 2017

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

References

(For references regarding interlacing roots, see A202605.)

Links

Table of n, a(n) for n=1..48.

Eric Weisstein's World of Mathematics, Detour Polynomial

Eric Weisstein's World of Mathematics, Distance Polynomial

Eric Weisstein's World of Mathematics, Path Graph

Index entries for sequences related to Chebyshev polynomials.

Formula

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

Example

The array T (a table if row n=0 is by convention put to 0) begins:
n\k     0      1      2       3       4       5      6      7     8    9  10 ...
(0:     0)
1:      0     -1
2:     -1      0      1
3:      4      6      0      -1
4:    -12    -32    -20       0       1
5:     32    120    140      50       0      -1
6:    -80   -384   -648    -448    -105       0      1
7:    192   1120   2464    2520    1176     196      0     -1
8:   -448  -3072  -8320  -11264   -7920   -2688   -336      0     1
9:   1024   8064  25920   43680   41184   21384   5544    540     0   -1
10: -2304 -20480 -76160 -153600 -182000 -128128 -51480 -10560  -825    0   1
... reformatted and extended. - Wolfdieter Lang, Feb 02 2018

Mathematica

(* begin*)
f[i_, j_] := Abs[i - j];
m[n_] := Table[f[i, j], {i, 1, n}, {j, 1, n}]
TableForm[m[6]] (* 6x6 principal submatrix *)
Flatten[Table[f[i, n + 1 - i],
{n, 1, 12}, {i, 1, n}]]  (* A049581 *)
p[n_] := CharacteristicPolynomial[m[n], x];
c[n_] := CoefficientList[p[n], x]
TableForm[Flatten[Table[p[n], {n, 1, 10}]]]
Table[c[n], {n, 1, 12}]
Flatten[%]    (* A203993 *)
TableForm[Table[c[n], {n, 1, 10}]]
(* end *)
CoefficientList[Table[CharacteristicPolynomial[SparseArray[{i_, j_} :> Abs[i - j], n], x], {n, 10}], x] //Flatten (* Eric W. Weisstein, Apr 07 2017 *)
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 *)
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 *)
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 *)

Crossrefs

Cf. A049310, A049581, A053120, A085750 (column k=0, Det(M_n)), A166445(n-1) (alternating row sums), A202605.

Sequence in context: A184187 A106145 A198372 * A204017 A021960 A096256

Adjacent sequences: A203990 A203991 A203992 * A203994 A203995 A203996

Keyword

tabl,sign,easy

Author

Clark Kimberling, Jan 09 2012

Status

approved