A202605 -id:A202605 - cp's OEIS frontend

A204021 Triangle read by rows: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of min(2i-1,2j-1) (A157454).

1, 1, -1, 2, -4, 1, 4, -12, 9, -1, 8, -32, 40, -16, 1, 16, -80, 140, -100, 25, -1, 32, -192, 432, -448, 210, -36, 1, 64, -448, 1232, -1680, 1176, -392, 49, -1, 128, -1024, 3328, -5632, 5280, -2688, 672, -64, 1, 256, -2304, 8640, -17472, 20592

Offset: 0

Clark Kimberling, Jan 11 2012

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 and A204016 for guides to related sequences.
a(0)=1 by convention. - Philippe Deléham, Nov 17 2013
The n roots of the n-th polynomial are 1/(1+cos((2*k-1)*Pi/(2*n))) for k = 1..n. See my pdf in the link section for the proof. - Jianing Song, Dec 01 2023

			Top of the triangle:
  1
  1....-1
  2....-4.....1
  4....-12....9....-1
  8....-32....40...-16....1
  16...-80....140..-100...25....-1
  32...-192...432..-448...210...-36....1
  ...
-448=2*(-100)-2*140-(-32). - _Philippe Deléham_, Nov 17 2013

(For references regarding interlacing roots, see A202605.)

Eric Weisstein's World of Mathematics, Morgan-Voyce polynomials
Jianing Song, Roots of the characteristic polynomials

Cf. A157454, A202605, A204016. A085478, A211957.

f[i_, j_] := Min[2 i - 1, 2 j - 1];
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, 15}, {i, 1, n}]]   (* A157454 *)
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[%]                  (* A204021 *)
TableForm[Table[c[n], {n, 1, 10}]]

From Peter Bala, May 01 2012: (Start)
The triangle appears to be a signed version of the row reverse of A211957.
If true, then for 0 <= k <= n-1, T(n,k) = (-1)^k*n/(n-k)*2^(n-k-1)*binomial(2*n-k-1,k) and Sum_{k = 0..n} T(n,k)*x^(n-k) = 1/2*(-1)^n*(b(2*n,-2*x) + 1)/b(n,-2*x), where b(n,x) := Sum_{k = 0..n} binomial(n+k,2*k)*x^k are the Morgan-Voyce polynomials of A085478.
Conjectural o.g.f.: t*(1-x-x^2*t)/(1-2*t*(1-x)+t^2*x^2) = (1-x)*t + (2-4*x+x^2)*t^2 + .... (End)
T(n,k)=2*T(n-1,k)-2*T(n-1,k-1)-T(n-2,k-2), T(0,0)=T(1,0)=1, T(1,1)=-1, T(n,k)=0 of k<0 or if k>n. - Philippe Deléham, Nov 17 2013

A202671 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of the symmetric matrix A202670 based on A000290 (the squares); by antidiagonals.

Original entry on oeis.org

1, -1, 1, -18, 1, 1, -84, 116, -1, 1, -214, 1707, -470, 1, 1, -408, 9430, -17896, 1449, -1, 1, -666, 31877, -196046, 124782, -3724, 1, 1, -988, 81720, -1120768, 2530948, -656400, 8400, -1, 1, -1374, 175727, -4386774, 23536143

Offset: 1

Clark Kimberling, Dec 22 2011

Let p(n)=p(n,x) be the characteristic polynomial of the n-th principal submatrix. The zeros of p(n) are positive, and they interlace the zeros of p(n+1).

			The 1st principal submatrix (ps) of A202670 is {{1}} (using Mathematica matrix notation), with p(1)=1-x and zero-set {1}.
...
The 2nd ps is {{1,4},{4,17}}, with p(2)=1-18x+x^2 and zero-set {0.556..., 17.944...}.
...
The 3rd ps is {{1,4,9},{4,17,40},{9,40,98}}, with p(3)=1-84x+116x^2-x^3 and zero-set {0.012..., 0.716..., 115.271...}.
...
Top of the array:
1...-1
1...-18..  ..1
1...-84... 116.....-1
1...-214...1707..-470...1

S.-G. Hwang, Cauchy's interlace theorem for eigenvalues of Hermitian matrices, American Mathematical Monthly 111 (2004) 157-159.
A. Mercer and P. Mercer, Cauchy's interlace theorem and lower bounds for the spectral radius, International Journal of Mathematics and Mathematical Sciences 23, no. 8 (2000) 563-566.

Cf. A202670, A000290, A202605 (the Fibonacci case).

f[k_] := k^2
U[n_] := NestList[Most[Prepend[#, 0]] &, #, Length[#] - 1] &[Table[f[k], {k, 1, n}]];
L[n_] := Transpose[U[n]];
F[n_] := CharacteristicPolynomial[L[n].U[n], x];
c[n_] := CoefficientList[F[n], x]
TableForm[Flatten[Table[F[n], {n, 1, 10}]]]
Table[c[n], {n, 1, 12}]
Flatten[%]
TableForm[Table[c[n], {n, 1, 10}]]

A203950 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of A203949.

Original entry on oeis.org

1, -1, 1, -3, 1, 1, -6, 5, -1, 1, -13, 18, -8, 1, 1, -24, 52, -40, 12, -1, 1, -39, 131, -155, 78, -16, 1, 1, -58, 291, -508, 391, -138, 21, -1, 1, -81, 584, -1410, 1548, -840, 225, -27, 1, 1, -108, 1078, -3448, 5151, -4016, 1658, -348, 33, -1, 1

Offset: 1

Clark Kimberling, Jan 08 2012

Let p(n)=p(n,x) be the characteristic polynomial of the n-th principal submatrix. The zeros of p(n) are positive, and they interlace the zeros of p(n+1). See A202605 for a guide to related sequences.

			Top of the array:
1...-1
1...-3....1
1...-6....5....-1
1...-13....18...-8....1
1...-24...52...-40...12...-1

(For references regarding interlacing roots, see A202605.)

Cf. A203949, A202605.

t = {1, 1, 0}; t1 = Flatten[{t, t, t, t, t, t, t, t, t}];
f[k_] := t1[[k]];
U[n_] :=
  NestList[Most[Prepend[#, 0]] &, #, Length[#] - 1] &[
   Table[f[k], {k, 1, n}]];
L[n_] := Transpose[U[n]];
p[n_] := CharacteristicPolynomial[L[n].U[n], x];
c[n_] := CoefficientList[p[n], x]
TableForm[Flatten[Table[p[n], {n, 1, 10}]]]
Table[c[n], {n, 1, 12}]  (* A203950 *)
Flatten[%]
TableForm[Table[c[n], {n, 1, 10}]]

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

Original entry on oeis.org

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

Clark Kimberling, Jan 09 2012

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

			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

(For references regarding interlacing roots, see A202605.)

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.

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

(* 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 *)

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

A203995 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of min{i-j+1,j-i+1} (A203994).

Original entry on oeis.org

1, -1, 1, -2, 1, 0, -2, 3, -1, -4, 8, 0, -4, 1, -16, 56, -56, 10, 5, -1, -48, 224, -360, 224, -35, -6, 1, -128, 736, -1584, 1560, -672, 84, 7, -1, -320, 2176, -5824, 7744, -5280, 1680, -168, -8, 1, -768, 6016, -19200, 32032, -29744

Offset: 1

Clark Kimberling, Jan 09 2012

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.

			Top of the array:
 1...-1
 1...-2....1
 0...-2....3...-1
-4....8....0...-4....1

(For references regarding interlacing roots, see A202605.)

Cf. A203994, A202605.

f[i_, j_] := Min[i - j + 1, j - i + 1];
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}]]  (* A203994 *)
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[%]    (* A203995 *)
TableForm[Table[c[n], {n, 1, 10}]]

A202672 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of the symmetric matrix A087062 based on (1,1,1,1,...); by antidiagonals.

Original entry on oeis.org

1, -1, 1, -3, 1, 1, -5, 6, -1, 1, -7, 15, -10, 1, 1, -9, 28, -35, 15, -1, 1, -11, 45, -84, 70, -21, 1, 1, -13, 66, -165, 210, -126, 28, -1, 1, -15, 91, -286, 495, -462, 210, -36, 1, 1, -17, 120, -455, 1001, -1287, 924, -330, 45, -1, 1, -19, 153

Offset: 1

Clark Kimberling, Dec 22 2011

Let p(n)=p(n,x) be the characteristic polynomial of the n-th principal submatrix of A087062. The zeros of p(n) are positive, and they interlace the zeros of p(n+1).

Closely related to A076756; however, for example, successive rows of A076756 are (1,-3,1), (-1,5,-6,1), compared to rows (1,-3,1), (1,-5,6,-1) of A202672.

			The 1st principal submatrix (ps) of A087062 is {{1}} (using Mathematica matrix notation), with p(1)=1-x and zero-set {1}.
...
The 2nd ps is {{1,1},{1,2}}, with p(2)=1-3x+x^2 and zero-set {0.381..., 2.618...}.
...
The 3rd ps is {{1,1,1},{1,2,2},{1,2,3}}, with p(3)=1-5x+6x^2-x^3 and zero-set {0.283..., 0.426..., 8.290...}.
...
Top of the array:
1...-1
1...-3....1
1...-5....6....-1
1...-7...15...-10....1
1...-9...28...-35...15...-1

S.-G. Hwang, Cauchy's interlace theorem for eigenvalues of Hermitian matrices, American Mathematical Monthly 111 (2004) 157-159.
A. Mercer and P. Mercer, Cauchy's interlace theorem and lower bounds for the spectral radius, International Journal of Mathematics and Mathematical Sciences 23, no. 8 (2000) 563-566.

Cf. A087062, A202673 (based on n), A202671 (based on n^2), A202605 (based on Fibonacci numbers), A076756.

U[n_] := NestList[Most[Prepend[#, 0]] &, #, Length[#] - 1] &[Table[1, {k, 1, n}]];
L[n_] := Transpose[U[n]];
F[n_] := CharacteristicPolynomial[L[n].U[n], x];
c[n_] := CoefficientList[F[n], x]
TableForm[Flatten[Table[F[n], {n, 1, 10}]]]
Table[c[n], {n, 1, 12}]
Flatten[%]
TableForm[Table[c[n], {n, 1, 10}]]
Table[(F[k] /. x -> -2), {k, 1, 30}] (* A007583 *)
Table[(F[k] /. x -> 2), {k, 1, 30}]  (* A087168 *)

A202673 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of the symmetric matrix A115263 based on (1,2,3,4,...); by antidiagonals.

Original entry on oeis.org

1, -1, 1, -6, 1, 1, -12, 20, -1, 1, -18, 75, -50, 1, 1, -24, 166, -328, 105, -1, 1, -30, 293, -1050, 1134, -196, 1, 1, -36, 456, -2432, 5140, -3312, 336, -1, 1, -42, 655, -4690, 15471, -20814, 8514, -540, 1, 1, -48, 890, -8040, 36771, -80584

Offset: 1

Clark Kimberling, Dec 22 2011

Let p(n)=p(n,x) be the characteristic polynomial of the n-th principal submatrix of A115262 (when A115262 is formatted as a square matrix). The zeros of p(n) are positive, and they interlace the zeros of p(n+1).

			The 1st principal submatrix (ps) of A115263 is {{1}} (using Mathematica matrix notation), with p(1)=1-x and zero-set {1}.
...
The 2nd ps is {{1,2},{2,5}}, with p(2)=1-6x+x^2 and zero-set {0.171..., 5.828...}.
...
The 3rd ps is {{1,2,3},{2,5,8},{3,8,14}}, with p(3)=1-12x+20x^2-x^3 and zero-set {0.099..., 0.516..., 19.383...}.
...
Top of the array:
1....-1
1....-6.....1
1...-12....20.....-1
1...-18....75....-50....1
1...-24...166...-328..105..-1

S.-G. Hwang, Cauchy's interlace theorem for eigenvalues of Hermitian matrices, American Mathematical Monthly 111 (2004) 157-159.
A. Mercer and P. Mercer, Cauchy's interlace theorem and lower bounds for the spectral radius, International Journal of Mathematics and Mathematical Sciences 23, no. 8 (2000) 563-566.

Cf. A115262, A202671 (based on n^2), A202605 (based on Fibonacci numbers)

U[n_] := NestList[Most[Prepend[#, 0]] &, #, Length[#] - 1] &[Table[k, {k, 1, n}]];
L[n_] := Transpose[U[n]];
F[n_] := CharacteristicPolynomial[L[n].U[n], x];
c[n_] := CoefficientList[F[n], x]
TableForm[Flatten[Table[F[n], {n, 1, 10}]]]
Table[c[n], {n, 1, 12}]
Flatten[%]
TableForm[Table[c[n], {n, 1, 10}]]

A202677 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of the symmetric matrix A202676 based on (1,4,7,10,13,...); by antidiagonals.

Original entry on oeis.org

1, -1, 1, -18, 1, 1, -116, 84, -1, 1, -538, 1347, -250, 1, 1, -2256, 11566, -8216, 585, -1, 1, -9158, 75453, -118722, 35086, -1176, 1, 1, -36796, 426288, -1152432, 801084, -118656, 2128, -1, 1, -147378, 2214919, -8910538, 11175711, -4079622, 339762, -3564, 1, 1, -589736, 10915650

Offset: 1

Clark Kimberling, Dec 22 2011

Let p(n)=p(n,x) be the characteristic polynomial of the n-th principal submatrix. The zeros of p(n) are positive, and they interlace the zeros of p(n+1).

			The 1st principal submatrix (ps) of A202676 is {{1}} (using Mathematica matrix notation), with p(1)=1-x and zero-set {1}.
...
The 2nd ps is {{1,4},{4,17}}, with p(2)=1-18x+x^2 and zero-set {0.055..., 17.944...}.
...
The 3rd ps is {{1,4,7},{4,17,32},{7,32,66}}, with p(3)=1-116x+84x^2-x^3 and zero-set {0.008..., 1.395..., 82.595...}.
...
Top of the array:
1...-1
1...-18....1
1...-116...84.....-1
1...-538...1347...-250...1

S.-G. Hwang, Cauchy's interlace theorem for eigenvalues of Hermitian matrices, American Mathematical Monthly 111 (2004) 157-159.
A. Mercer and P. Mercer, Cauchy's interlace theorem and lower bounds for the spectral radius, International Journal of Mathematics and Mathematical Sciences 23, no. 8 (2000) 563-566.

Cf. A202676, A202677, A202605.

f[k_] := 3 k - 2
U[n_] := NestList[Most[Prepend[#, 0]] &, #, Length[#] - 1] &[Table[f[k], {k, 1, n}]];
L[n_] := Transpose[U[n]];
F[n_] := CharacteristicPolynomial[L[n].U[n], x];
c[n_] := CoefficientList[F[n], x]
TableForm[Flatten[Table[F[n], {n, 1, 10}]]]
Table[c[n], {n, 1, 12}]
Flatten[%]
TableForm[Table[c[n], {n, 1, 10}]]

A202678 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of the symmetric matrix A185957; by antidiagonals.

Original entry on oeis.org

1, -1, 1, -11, 1, 1, -30, 57, -1, 1, -50, 395, -203, 1, 1, -70, 1133, -3221, 574, -1, 1, -90, 2271, -15840, 19011, -1386, 1, 1, -110, 3809, -45980, 156892, -88729, 2982, -1, 1, -130, 5747, -101640, 660617, -1195097, 346295, -5874

Offset: 1

Clark Kimberling, Dec 22 2011

Let p(n)=p(n,x) be the characteristic polynomial of the n-th principal submatrix. The zeros of p(n) are positive, and they are interlace the zeros of p(n+1).

			The 1st principal submatrix (ps) of A185957 is {{1}} (using Mathematica matrix notation), with p(1)=1-x and zero-set {1}.
...
The 2nd ps is {{1,3},{3,10}}, with p(2)=1-11x+x^2 and zero-set {0.091..., 10.908...}.
...
The 3rd ps is {{1,3,6},{3,10,21},{6,21,46}}, with p(3)=1-30x+57x^2-x^3 and zero-set {0.035..., 0.495..., 56.469...}.
...
Top of the array:
1...-1
1...-11...1
1...-30...57....-1
1...-50...395...-203...1

S.-G. Hwang, Cauchy's interlace theorem for eigenvalues of Hermitian matrices, American Mathematical Monthly 111 (2004) 157-159.
A. Mercer and P. Mercer, Cauchy's interlace theorem and lower bounds for the spectral radius, International Journal of Mathematics and Mathematical Sciences 23, no. 8 (2000) 563-566.

Cf. A185957, A202605.

f[k_] := k (k + 1)/2
U[n_] :=  NestList[Most[Prepend[#, 0]] &, #, Length[#] - 1] &[Table[f[k], {k, 1, n}]];
L[n_] := Transpose[U[n]];
F[n_] := CharacteristicPolynomial[L[n].U[n], x];
c[n_] := CoefficientList[F[n], x]
TableForm[Flatten[Table[F[n], {n, 1, 10}]]]
Table[c[n], {n, 1, 12}]
Flatten[%]
TableForm[Table[c[n], {n, 1, 10}]]

A202868 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of the symmetric matrix A115216; by antidiagonals.

Original entry on oeis.org

1, -1, 1, -6, 1, 1, -11, 27, -1, 1, -16, 78, -112, 1, 1, -21, 154, -458, 453, -1, 1, -26, 255, -1164, 2431, -1818, 1, 1, -31, 381, -2355, 7635, -12141, 7279, -1, 1, -36, 532, -4156, 18390, -45660, 58260, -29124, 1, 1, -41, 708, -6692, 37646, -128190

Offset: 1

Clark Kimberling, Dec 26 2011

Let p(n)=p(n,x) be the characteristic polynomial of the n-th principal submatrix. The zeros of p(n) are positive, and they interlace the zeros of p(n+1).

			The 1st principal submatrix (ps) of A115216 is {{1}} (using Mathematica matrix notation), with p(1)=1-x and zero-set {1}.
...
The 2nd ps is {{1,2},{2,5}}, with p(2)=1-6x+x^2 and zero-set {0.171..., 5.828...}.
...
The 3rd ps is {{1,2,4},{2,5,10},{4,10,21}}, with p(3)=1-30x+57x^2-x^3 and zero-set {0.136..., 0.276..., 2.587...}.
...
Top of the array:
1...-1
1...-6....1
1...-11...27...-1
1...-16...78...-112...1

S.-G. Hwang, Cauchy's interlace theorem for eigenvalues of Hermitian matrices, American Mathematical Monthly 111 (2004) 157-159.
A. Mercer and P. Mercer, Cauchy's interlace theorem and lower bounds for the spectral radius, International Journal of Mathematics and Mathematical Sciences 23, no. 8 (2000) 563-566.

Cf. A115216, A202605.

f[k_] := 2^(k - 1);
U[n_] := NestList[Most[Prepend[#, 0]] &, #, Length[#] - 1] &[Table[f[k], {k, 1, n}]];
L[n_] := Transpose[U[n]];
F[n_] := CharacteristicPolynomial[L[n].U[n], x];
c[n_] := CoefficientList[F[n], x]
TableForm[Flatten[Table[F[n], {n, 1, 10}]]]
Table[c[n], {n, 1, 12}]
Flatten[%]  (* A202868 sequence *)
TableForm[Table[c[n], {n, 1, 10}]]  (* A202868 array *)
Table[(F[k] /. x -> -1), {k, 1, 30}]   (* A154626 *)
Table[(F[k] /. x -> 1), {k, 1, 30}]    (* A058922 *)

cp's OEIS Frontend

A204021 Triangle read by rows: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of min(2i-1,2j-1) (A157454).

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A202671 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of the symmetric matrix A202670 based on A000290 (the squares); by antidiagonals.

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A203950 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of A203949.

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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).

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A203995 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of min{i-j+1,j-i+1} (A203994).

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A202672 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of the symmetric matrix A087062 based on (1,1,1,1,...); by antidiagonals.

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A202673 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of the symmetric matrix A115263 based on (1,2,3,4,...); by antidiagonals.

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A202677 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of the symmetric matrix A202676 based on (1,4,7,10,13,...); by antidiagonals.

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