A202605 -id:A202605 - cp's OEIS frontend

A203005 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of A115255 (in square format); by antidiagonals.

1, -1, 1, -6, 1, 1, -15, 47, -1, 1, -40, 270, -488, 1, 1, -165, 1738, -5866, 5829, -1, 1, -1074, 15695, -80060, 156495, -74674, 1, 1, -9039, 181581, -1360515, 4552003, -5997165, 997295, -1, 1, -86700, 2566036, -28081556

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...-6....1
1...-15...47....-1
1...-40...270...-488...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. A115255, A202605.

f[k_] := Binomial[2 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}]]

A203948 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of A203947.

Original entry on oeis.org

1, -1, 1, -2, 1, 1, -4, 4, -1, 1, -7, 13, -7, 1, 1, -11, 35, -31, 10, -1, 1, -16, 74, -107, 61, -14, 1, 1, -22, 147, -308, 275, -111, 19, -1, 1, -29, 256, -763, 1001, -629, 186, -24, 1, 1, -37, 428, -1683, 3013, -2721, 1264, -291, 30, -1, 1, -46

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...-2....1
1...-4....4....-1
1...-7....13...-7....1
1...-11...35...-31...10...-1

(For references regarding interlacing roots, see A202605.)

Cf. A203947, A202605.

t = {1, 0, 1}; t1 = Flatten[{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}]
Flatten[%]
TableForm[Table[c[n], {n, 1, 10}]]

A203952 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, -2, 1, 1, -3, 3, -1, 1, -4, 6, -4, 1, 1, -6, 13, -13, 6, -1, 1, -8, 24, -34, 24, -8, 1, 1, -10, 39, -75, 75, -39, 10, -1, 1, -12, 58, -144, 195, -144, 58, -12, 1, 1, -14, 81, -250, 444, -459, 271, -89, 15, -1, 1, -16, 108, -400, 886

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. A203951, 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}]]

A203992 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of (A143182 in square format).

Original entry on oeis.org

1, -1, -3, -2, 1, 8, 14, 3, -1, -20, -56, -40, -4, 1, 48, 184, 224, 90, 5, -1, -112, -544, -936, -672, -175, -6, 1, 256, 1504, 3344, 3480, 1680, 308, 7, -1, -576, -3968, -10816, -14784, -10560, -3696, -504, -8, 1, 1280, 10112, 32640, 55328

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
-3... -1.... 1
 8.... 14... 3... -1
-20.. -56.. -40.. -4... 1

(For references regarding interlacing roots, see A202605.)

Cf. A143182, A202605.

f[i_, j_] := Max[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}]]  (* A143182 in square format *)
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[%]    (* A203992 *)
TableForm[Table[c[n], {n, 1, 10}]]

A204017 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of max{j mod i, i mod j} (A204016).

Original entry on oeis.org

0, -1, -1, 0, 1, 4, 6, 0, -1, -15, -38, -20, 0, 1, 56, 206, 184, 50, 0, -1, -185, -1072, -1357, -630, -105, 0, 1, 204, 5146, 9276, 6060, 1736, 196, 0, -1, 6209, -17334, -58470, -52452, -21102, -4116, -336, 0, 1, -112400, -67682, 293984

Offset: 1

Clark Kimberling, Jan 10 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 A204016 and A202605 for guides to related sequences.

			Top of the array:
 1... -1
-1.... 0.... 1
 4.... 6.... 0... -1
-15.. -38.. -20... 0... 1
 56... 206.. 184.. 50.. 0.. -1
...
The 1st principal submatrix (ps) of A204016 is {{1}} (using Mathematica matrix notation), with p(1)=1-x and zero-set {1}.
...
The 2nd ps is {{0,1},{1,0}}, with p(2)=-1+x^2 and zero-set {-1,1}.
...
The 3rd ps is {{0,1,1},{1,0,2},{1,2,0}}, with p(3)=4+6x-x^3 and zero-set {-2, -0.732...,2.732...}.
...
The 4th ps is {{0,1,1,1},{1,0,2,2},{1,2,0,3},{1,2,0,3}}, with p(4)=-15-38x-20x^2+x^4 and zero-set {-3, -1.714, -0.553, 5.268}.
...
The interlace property is illustrated for the last two zero-sets by this chain:
-3 < -2 < -1.7 < -0.7 < -0.5 < 2.7 < 5.2

(For references regarding interlacing roots, see A202605.)

Cf. A204016, A202605.

f[i_, j_] := Max[Mod[i, j], Mod[j, i]];
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, 12}, {i, 1, n}]]  (* A204016 *)
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[%]               (* A204017 *)
TableForm[Table[c[n], {n, 1, 10}]]

A204020 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of min(i,j)^2 (A106314).

Original entry on oeis.org

1, -1, 3, -5, 1, 15, -31, 14, -1, 105, -247, 157, -30, 1, 945, -2433, 1892, -553, 55, -1, 10395, -28653, 25573, -9620, 1554, -91, 1, 135135, -393279, 388810, -173773, 37550, -3738, 140, -1, 2027025, -6169455

Offset: 1

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.

Constant term of p(n,x) is A001147(n), and the coefficient of the linear term is A000330(n). - Enrique Pérez Herrero, Feb 20 2013

			Top of the array:
1.....-1
3.....-5.....1
15....-31....14....-1
105...-247...157...-30...1

(For references regarding interlacing roots, see A202605.)

Cf. A106314, A202605, A204016.

f[i_, j_] := Min[i^2, j^2];
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}]]   (* A106314 *)
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[%]                  (* A204020 *)
TableForm[Table[c[n], {n, 1, 10}]]

A204025 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of gcd(i,j) (A003989).

Original entry on oeis.org

1, -1, 1, -3, 1, 2, -8, 6, -1, 4, -20, 26, -10, 1, 16, -88, 134, -72, 15, -1, 32, -240, 496, -408, 143, -21, 1, 192, -1504, 3352, -3112, 1344, -284, 28, -1, 768, -6400, 16320, -18496, 10508, -3108, 480, -36, 1, 4608, -39936, 109952

Offset: 1

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.

			Top of the array:
  1,  -1;
  1,  -3,   1;
  2,  -8,   6,  -1;
  4, -20,  26, -10,   1;

(For references regarding interlacing roots, see A202605.)

Cf. A003989, A202605, A204016.

f[i_, j_] := GCD[i, j]
m[n_] := Table[f[i, j], {i, 1, n}, {j, 1, n}]
TableForm[m[6]] (* 6 X 6 principal submatrix *)
Flatten[Table[f[i, n + 1 - i],
{n, 1, 15}, {i, 1, n}]]    (* A003989 *)
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[%]                 (* A204025 *)
TableForm[Table[c[n], {n, 1, 10}]]

A204122 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of f(i,j) = gcd(2^(i-1), 2^(j-1)) (A144464).

Original entry on oeis.org

1, -1, 1, -3, 1, 2, -8, 7, -1, 8, -36, 43, -15, 1, 64, -304, 414, -198, 31, -1, 1024, -4992, 7224, -3960, 849, -63, 1, 32768, -161792, 241088, -140864, 34674, -3516, 127, -1, 2097152, -10420224, 15752192, -9492480, 2493640, -290412

Offset: 1

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.

			Top of the array:
  1,  -1;
  1,  -3,   1;
  2,  -8,   7,  -1;
  8, -36,  43, -15,   1;

(For references regarding interlacing roots, see A202605.)

Cf. A144464, A202605, A204016.

f[i_, j_] := GCD[2^(i - 1), 2^(j - 1)];
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}]] (* A144464 *)
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[%]              (* A204122 *)
TableForm[Table[c[n], {n, 1, 10}]]

A204168 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of (i+j), as in A003057.

Original entry on oeis.org

2, -1, -1, -6, 1, 0, 6, 12, -1, 0, 0, -20, -20, 1, 0, 0, 0, 50, 30, -1, 0, 0, 0, 0, -105, -42, 1, 0, 0, 0, 0, 0, 196, 56, -1, 0, 0, 0, 0, 0, 0, -336, -72, 1, 0, 0, 0, 0, 0, 0, 0, 540, 90, -1, 0, 0, 0, 0, 0, 0, 0, 0, -825, -110, 1

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:
 2....-1
-1....-6.....1
 0.....6.....12....-1
 0.....0....-20....-20...1

(For references regarding interlacing roots, see A202605.)

Cf. A003057, A202605, A204016.

f[i_, j_] := i + j;
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}]]  (* A003057 *)
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[%]                 (* A204168 *)
TableForm[Table[c[n], {n, 1, 10}]]

A204169 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of (i+j-1), as in A002024.

Original entry on oeis.org

1, -1, -1, -4, 1, 0, 6, 9, -1, 0, 0, -20, -16, 1, 0, 0, 0, 50, 25, -1, 0, 0, 0, 0, -105, -36, 1, 0, 0, 0, 0, 0, 196, 49, -1, 0, 0, 0, 0, 0, 0, -336, -64, 1, 0, 0, 0, 0, 0, 0, 0, 540, 81, -1, 0, 0, 0, 0, 0, 0, 0, 0, -825, -100, 1

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:
2....-1
-1....-4.....1
0.....6.....9....-1
0.....0....-20...-16...1

(For references regarding interlacing roots, see A202605.)

Cf. A002024, A202605, A204016.

f[i_, j_] := i + j - 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}]]  (* A002024 *)
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[%]                (* A204169 *)
TableForm[Table[c[n], {n, 1, 10}]]

cp's OEIS Frontend

A203005 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of A115255 (in square format); by antidiagonals.

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A203948 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of A203947.

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A203952 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of A203949.

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A203992 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of (A143182 in square format).

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A204017 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of max{j mod i, i mod j} (A204016).

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A204020 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of min(i,j)^2 (A106314).

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A204025 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of gcd(i,j) (A003989).

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Mathematica

A204122 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of f(i,j) = gcd(2^(i-1), 2^(j-1)) (A144464).

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