A202605 -id:A202605 - cp's OEIS frontend

A204163 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of (floor[(i+1)/2] if i=j and = 0 otherwise), as in A204162.

1, -1, 0, -2, 1, 0, -2, 4, -1, 0, -2, 7, -6, 1, 0, -4, 17, -21, 9, -1, 0, -8, 40, -64, 43, -12, 1, 0, -24, 132, -244, 206, -85, 16, -1, 0, -72, 432, -904, 913, -492, 142, -20, 1, 0, -288, 1836, -4180, 4749, -3025, 1118, -234, 25, -1, 0

Offset: 1

Clark Kimberling, Jan 12 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.

			Top of the array:
1....-1
0....-2....1
0....-2....4....-1
0....-4....17...-21...9...1

(For references regarding interlacing roots, see A202605.)

Cf. A204162, A202605, A204016.

f[i_, j_] := 1; f[i_, i_] := Floor[(i + 1)/2];
m[n_] := Table[f[i, j], {i, 1, n}, {j, 1, n}]
TableForm[m[8]] (* 8x8 principal submatrix *)
Flatten[Table[f[i, n + 1 - i],
  {n, 1, 15}, {i, 1, n}]]  (* A204162 *)
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[%]                (* A204163 *)
TableForm[Table[c[n], {n, 1, 10}]]

A204165 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of floor[(i+j)/2], as in A204164.

Original entry on oeis.org

1, -1, 1, -3, 1, -1, -2, 6, -1, 0, 4, 4, -10, 1, 0, 0, -15, -4, 15, -1, 0, 0, 0, 36, 3, -21, 1, 0, 0, 0, 0, -84, 4, 28, -1, 0, 0, 0, 0, 0, 160, -16, -36, 1, 0, 0, 0, 0, 0, 0, -300, 40, 45, -1, 0, 0, 0, 0, 0, 0, 0, 500, -75, -55, 1, 0, 0, 0, 0, 0, 0, 0, 0, -825, 130

Offset: 1

Clark Kimberling, Jan 12 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.

			Top of the array:
 1....-1
 1....-3.....1
-1....-2.....6....-1
 0.....4.....4....-10...1

(For references regarding interlacing roots, see A202605.)

Cf. A204164, A202605, A204016.

f[i_, j_] := Floor[(i + j)/2];
m[n_] := Table[f[i, j], {i, 1, n}, {j, 1, n}]
TableForm[m[8]] (* 8x8 principal submatrix *)
Flatten[Table[f[i, n + 1 - i],
  {n, 1, 15}, {i, 1, n}]]  (* A204164 *)
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[%]                 (* A204165 *)
TableForm[Table[c[n], {n, 1, 10}]]

A204167 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of ceiling((i+j)/2), as in A204166.

Original entry on oeis.org

1, -1, -2, -3, 1, 1, 6, 6, -1, 0, -4, -16, -10, 1, 0, 0, 15, 32, 15, -1, 0, 0, 0, -36, -60, -21, 1, 0, 0, 0, 0, 84, 100, 28, -1, 0, 0, 0, 0, 0, -160, -160, -36, 1, 0, 0, 0, 0, 0, 0, 300, 240, 45, -1, 0, 0, 0, 0, 0, 0, 0, -500, -350

Offset: 1

Clark Kimberling, Jan 12 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.

			Top of the array:
   1,  -1
  -2,  -3,   1
   1,   6,   6,  -1
   0,  -4, -16, -10,   1

(For references regarding interlacing roots, see A202605.)

Cf. A204166, A202605, A204016.

f[i_, j_] := Ceiling[(i + j)/2];
m[n_] := Table[f[i, j], {i, 1, n}, {j, 1, n}]
TableForm[m[8]] (* 8 X 8 principal submatrix *)
Flatten[Table[f[i, n + 1 - i],
  {n, 1, 15}, {i, 1, n}]]  (* A204166 *)
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[%]                 (* A204167 *)
TableForm[Table[c[n], {n, 1, 10}]]

Definition corrected by Georg Fischer, Nov 29 2021

A204172 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of (f(i,j)), where f(i,j)=(1 if max(i,j) is odd, and 0 otherwise) as in A204171.

Original entry on oeis.org

1, -1, 0, -1, 1, -1, 1, 2, -1, 0, 1, -1, -2, 1, 1, -1, -4, 3, 3, -1, 0, -1, 1, 4, -3, -3, 1, -1, 1, 6, -5, -10, 6, 4, -1, 0, 1, -1, -6, 5, 10, -6, -4, 1, 1, -1, -8, 7, 21, -15, -20, 10, 5, -1, 0, -1, 1, 8, -7, -21, 15, 20, -10, -5, 1, -1, 1, 10, -9, -36, 28, 56

Offset: 1

Clark Kimberling, Jan 12 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.

This sequence uses the characteristic polynomial defined as det(A - x I), rather than det(x I - A), so the last term in row n is (-1)^n. - Robert Israel, Feb 10 2023

			Top of the array:
   1, -1;
   0, -1,  1;
  -1,  1,  2, -1;
   0,  1, -1, -2, 1;

(For references regarding interlacing roots, see A202605.)

Robert Israel, Table of n, a(n) for n = 1..10010 (rows 1 to 140, flattened)

Cf. A204171, A202605, A204016.

for n from 1 to 20 do
  P:= (-1)^n * LinearAlgebra:-CharacteristicPolynomial(Matrix(n,n,(i,j) -> max(i,j) mod 2),x):
  print(seq(coeff(P,x,i),i=0..n));
od: # Robert Israel, Feb 10 2023

f[i_, j_] := If[Mod[Max[i, j], 2] == 1, 1, 0]
m[n_] := Table[f[i, j], {i, 1, n}, {j, 1, n}]
TableForm[m[8]] (* 8x8 principal submatrix *)
Flatten[Table[f[i, n + 1 - i],
  {n, 1, 15}, {i, 1, n}]]  (* A204171 *)
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[%]                 (* A204172 *)
TableForm[Table[c[n], {n, 1, 10}]]

(Empirical) T(m,k) = [x^m y^(k-1)] y*(1-x*y)*(1-x+x^3*y^2)/(1+y^2-2*x^2*y^2+x^4*y^4). - Robert Israel, Feb 10 2023

A204184 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of (f(i,j)), where f(i,1)=f(1,j)=1, f(i,i)=(-1)^(i-1); f(i,j)=0 otherwise; as in A204181.

Original entry on oeis.org

1, -1, -2, 0, 1, -1, 3, 1, -1, 2, -2, -5, 0, 1, 1, -5, -2, 6, 1, -1, -2, 4, 9, -4, -8, 0, 1, -1, 7, 3, -15, -3, 9, 1, -1, 2, -6, -13, 12, 21, -6, -11, 0, 1, 1, -9, -4, 28, 6, -30, -4, 12, 1, -1, -2, 8, 17, -24, -40, 24, 38, -8, -14, 0, 1, -1, 11, 5

Offset: 1

Clark Kimberling, Jan 12 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.

			Top of the array:
 1..-1
 2...0...1
-1...3...1..-1
 2..-2..-5...0..1

(For references regarding interlacing roots, see A202605.)

Cf. A204183, A202605, A204016.

f[i_, j_] := 0; f[1, j_] := 1; f[i_, 1] := 1;
f[i_, i_] := (-1)^(i - 1);
m[n_] := Table[f[i, j], {i, 1, n}, {j, 1, n}]
TableForm[m[8]] (* 8x8 principal submatrix *)
Flatten[Table[f[i, n + 1 - i],
  {n, 1, 15}, {i, 1, n}]]  (* A204183 *)
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[%]                 (* A204184 *)
TableForm[Table[c[n], {n, 1, 10}]]

A202675 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of the symmetric matrix A202674 based on (1,3,5,7,9,...); by antidiagonals.

Original entry on oeis.org

1, -1, 1, -11, 1, 1, -37, 46, -1, 1, -79, 367, -130, 1, 1, -137, 1444, -2083, 295, -1, 1, -211, 4013, -13820, 8518, -581, 1, 1, -301, 9066, -58277, 89402, -27966, 1036, -1, 1, -407, 17851, -186166, 548591, -442118, 78354

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 A202674 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 {{1,3,5},{3,10,18},{5,18,35}}, with p(3)=1-37x+46x^2-x^3 and zero-set {0.012..., 0.716..., 115.271...}.
...
Top of the array:
1....-1
1...-11.....1
1...-37....46.....-1
1...-79...367...-130...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. A202674, 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[%]
TableForm[Table[c[n], {n, 1, 10}]]

A202872 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of the symmetric matrix A202871; by antidiagonals.

Original entry on oeis.org

1, -1, 1, -11, 1, 1, -46, 37, -1, 1, -181, 298, -112, 1, 1, -716, 1784, -1368, 308, -1, 1, -2851, 9495, -11119, 5286, -828, 1, 1, -11386, 47431, -74940, 55235, -18546, 2189, -1, 1, -45521, 227592, -453206, 455080, -239360, 61185, -5759

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 A202871 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 {{1,3,4},{3,10,15},{4,15,26}}, with p(3)=1-46x+37x^2-x^3 and zero-set {0.022..., 1.265..., 35.712...}.
...
Top of the array:
1...-1
1...-11....1
1...-46....37....-1
1...-181...298...-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. A202871, A202605.

f[k_] := LucasL[k];
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}]]

A202877 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of the symmetric matrix A202875; by antidiagonals.

Original entry on oeis.org

1, -1, 1, -6, 1, 1, -11, 27, -1, 1, -17, 84, -97, 1, 1, -23, 177, -497, 311, -1, 1, -29, 306, -1405, 2546, -925, 1, 1, -35, 471, -3034, 9375, -11628, 2628, -1, 1, -41, 672, -5599, 24817, -55080, 48875, -7247, 1, 1, -47, 909, -9316, 54164

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).

			Top of the array:
1...-1
1...-6....1
1...-11...27...-1
1...-17...84...-97...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. A202876, A202605.

f[k_] := -1 + Fibonacci[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}]]

A202971 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of the symmetric matrix A202970; by antidiagonals.

Original entry on oeis.org

1, -1, 1, -11, 1, 1, -30, 57, -1, 1, -53, 338, -224, 1, 1, -80, 992, -2600, 752, -1, 1, -111, 2171, -11803, 15614, -2304, 1, 1, -146, 4039, -35908, 105335, -79786, 6665, -1, 1, -185, 6776, -87154, 434244, -770624, 362449, -18595

Offset: 1

Clark Kimberling, Dec 27 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).

			Top of the array:
1...-1
1...-11...1
1...-30...57....-1
1...-53...338...-224...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. A202970, A202605.

f[k_] := -2 + Fibonacci[k + 3]
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}]]

A203002 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of the symmetric matrix A203001; by antidiagonals.

Original entry on oeis.org

1, -1, 1, -3, 1, 1, -14, 21, -1, 1, -29, 162, -120, 1, 1, -48, 540, -1736, 844, -1, 1, -71, 1267, -8091, 17022, -5664, 1, 1, -98, 2475, -24908, 105503, -158690, 39045, -1, 1, -129, 4312, -60994, 408508, -1250056, 1416673

Offset: 1

Clark Kimberling, Dec 27 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).

			Top of the array:
1...-1
1...-3....1
1...-14...21....-1
1...-29...162...-120...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. A203001, A202605.

f[k_] := Fibonacci[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}]]

cp's OEIS Frontend

A204163 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of (floor[(i+1)/2] if i=j and = 0 otherwise), as in A204162.

Views

Author

Keywords

Comments

Examples

References

Crossrefs

Programs

Mathematica

A204165 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of floor[(i+j)/2], as in A204164.

Views

Author

Keywords

Comments

Examples

References

Crossrefs

Programs

Mathematica

A204167 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of ceiling((i+j)/2), as in A204166.

Views

Author

Keywords

Comments

Examples

References

Crossrefs

Programs

Mathematica

Extensions

A204172 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of (f(i,j)), where f(i,j)=(1 if max(i,j) is odd, and 0 otherwise) as in A204171.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A204184 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of (f(i,j)), where f(i,1)=f(1,j)=1, f(i,i)=(-1)^(i-1); f(i,j)=0 otherwise; as in A204181.

Views

Author

Keywords

Comments

Examples

References

Crossrefs

Programs

Mathematica

A202675 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of the symmetric matrix A202674 based on (1,3,5,7,9,...); by antidiagonals.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A202872 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of the symmetric matrix A202871; by antidiagonals.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A202877 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of the symmetric matrix A202875; by antidiagonals.

Views

Author

Keywords

Comments