A202605 -id:A202605 - cp's OEIS frontend

A202870 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of the symmetric matrix A202869; by antidiagonals.

1, -1, 1, -11, 1, 1, -46, 37, -1, 1, -162, 299, -99, 1, 1, -567, 1675, -1324, 225, -1, 1, -1872, 8316, -11315, 5292, -432, 1, 1, -5881, 40254, -79457, 60782, -16458, 760, -1, 1, -17990, 182413, -490520, 543130, -260498, 45424, -1232

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 A202869 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...-162...299...-99...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. A202869, A202605.

f[k_] := Floor[k*GoldenRatio];
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[%]  (* A202870 as a sequence *)
TableForm[Table[c[n], {n, 1, 10}]]  (* A202870 as a matrix *)

A202875 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of the symmetric matrix A202874; by antidiagonals.

Original entry on oeis.org

1, -1, 1, -6, 1, 1, -12, 20, -1, 1, -19, 69, -59, 1, 1, -27, 159, -303, 162, -1, 1, -36, 302, -943, 1149, -434, 1, 1, -46, 511, -2284, 4599, -3991, 1147, -1, 1, -57, 800, -4743, 13733, -19785, 13090, -3016, 1, 1, -69, 1184, -8867, 34141, -70945

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...-12...20...-1
1...-19...69...-59...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. A202874, A202605.

f[k_] := Fibonacci[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}]]

A203004 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of the symmetric matrix A203003; by antidiagonals.

Original entry on oeis.org

1, -1, 1, -18, 1, 1, -84, 116, -1, 1, -439, 1221, -839, 1, 1, -2475, 10435, -13855, 5658, -1, 1, -14312, 81690, -165715, 138669, -39038, 1, 1, -83270, 601411, -1661956, 2164099, -1292751, 266899, -1, 1, -485157

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...-18....1
1...-84....116....-1
1...-439...1221...-839...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. A203003, A202605.

f[k_] := Fibonacci[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}]]

A203906 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of A203905.

Original entry on oeis.org

1, -1, 1, -2, 1, 1, -4, 4, -1, 1, -6, 11, -6, 1, 1, -8, 22, -24, 9, -1, 1, -10, 37, -62, 46, -12, 1, 1, -12, 56, -128, 148, -80, 16, -1, 1, -14, 79, -230, 367, -314, 130, -20, 1, 1, -16, 106, -376, 771, -920, 610, -200, 25, -1, 1, -18, 137

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.

If we omit the main diagonal of this array and ignore the signs of the entries then the resulting array, reading the rows in reverse order, appears to equal the Riordan array (1/((1 + x)*(1 - x)^3), x/(1 - x)^2), whose generating function begins 1 + (2 + t)*x + (4 + 4*t + t^2)*x^2 + (6 + 11*t + 6*t^2 + t^3)*x^3 + (9 + 24*t + 22*t^2 + 8*t^3 + t^4)*x^4 + .... - Peter Bala, Sep 17 2019

			Top of the array:
1...-1
1...-2....1
1...-4....4...-1
1...-6...11...-6....1
1...-8...22...-24...9...-1

(For references regarding interlacing roots, see A202605.)

Cf. A202605, A166516.

t = {1, 0}; t1 = Flatten[{t, 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}]
Flatten[%]                         (* A203906 *)
TableForm[Table[c[n], {n, 1, 10}]]
Table[p[n] /. x -> -1, {n, 1, 16}] (* A166516 *)

A203946 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of A203945.

Original entry on oeis.org

1, -1, 1, -2, 1, 1, -3, 3, -1, 1, -5, 8, -5, 1, 1, -7, 17, -17, 7, -1, 1, -9, 30, -45, 30, -9, 1, 1, -11, 47, -98, 103, -52, 12, -1, 1, -13, 68, -183, 269, -212, 83, -15, 1, 1, -15, 93, -308, 588, -651, 399, -123, 18, -1, 1, -17, 122, -481, 1136

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...-3....3....-1
1...-5....8....-5....1
1...-7....17...-17...7...-1

(For references regarding interlacing roots, see A202605.)

Cf. A203945, A202605.

t = {1, 0, 0}; t1 = Flatten[{t, 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}]
Flatten[%]   (* A203946 *)
TableForm[Table[c[n], {n, 1, 10}]]

A203954 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of A203953.

Original entry on oeis.org

1, -1, 1, -6, 1, 1, -20, 12, -1, 1, -70, 75, -22, 1, 1, -264, 406, -200, 33, -1, 1, -1034, 2085, -1470, 430, -48, 1, 1, -4108, 10296, -9600, 4116, -816, 64, -1, 1, -16398, 49231, -57574, 33135, -9786, 1410, -84, 1, 1, -65552, 229482

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...-6.....1
1...-20....12....-1
1...-70....75....-22....1
1...-264...406...-200...33...-1

(For references regarding interlacing roots, see A202605.)

Cf. A203953, A202605.

t = {1, 2}; t1 = Flatten[{t, 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}]
Flatten[%]  (* A203954 *)
TableForm[Table[c[n], {n, 1, 10}]]

A203956 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of A203955.

Original entry on oeis.org

1, -1, 1, -6, 1, 1, -12, 20, -1, 1, -27, 165, -35, 1, 1, -123, 1255, -511, 54, -1, 1, -300, 9266, -6003, 1197, -82, 1, 1, -558, 77523, -71564, 20779, -2463, 111, -1, 1, -2841, 688624, -817771, 315489, -54393, 4386, -144, 1, 1, -9093

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...-6....1
1...-12....20....-1
1...-27....165...-35....1
1...-123...1255..-511...54...-1

(For references regarding interlacing roots, see A202605.)

Cf. A203955, A202605.

t = {1, 2, 3}; 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}]
Flatten[%]  (* A203956 *)
TableForm[Table[c[n], {n, 1, 10}]]

A203989 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of {max(i,j)} (A051125).

Original entry on oeis.org

1, -1, -2, -3, 1, 3, 11, 6, -1, -4, -23, -35, -10, 1, 5, 39, 98, 85, 15, -1, -6, -59, -207, -308, -175, -21, 1, 7, 83, 374, 795, 798, 322, 28, -1, -8, -111, -611, -1694, -2475, -1806, -546, -36, 1, 9, 143, 930, 3185, 6149, 6633, 3696, 870, 45

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.

The characteristic polynomial seems be the recurrence relation given by p(n,x) = -x * p(n-1,x) + n * (-1)^(n-1) * sum_{i=0..n-1} x^i * binomial(2n-i-2,i). - Enrique Pérez Herrero, Jan 29 2013

			Top of the array:
1... -1
-2... -3.... 1
3.... 11... 6... -1
-4... -23.. -35.. -10...1
5.... 39... 98... 85...15.. -1

(For references regarding interlacing roots, see A202605.)

Cf. A051125, A202605.

Cf. A181983, A076756.

f[i_, j_] := Max[i, j];
m[n_] := Table[f[i, j], {i, 1, n}, {j, 1, n}]
TableForm[m[6]] (* 6th principal submatrix *)
Flatten[Table[f[i, n + 1 - i],
{n, 1, 12}, {i, 1, n}]]  (* A051125 *)
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[%]              (* A203989 *)
TableForm[Table[c[n], {n, 1, 10}]]

A203991 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of {(i+j)*min(i,j)} (A203990).

Original entry on oeis.org

2, -1, 7, -10, 1, 38, -71, 28, -1, 281, -610, 357, -60, 1, 2634, -6329, 4620, -1253, 110, -1, 29919, -77530, 65613, -23348, 3514, -182, 1, 399342, -1098271, 1036044, -442349, 90800, -8442, 280, -1, 6125265

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:
2.... -1
7.... -10... 1
38... -71... 28... -1
281.. -610.. 357.. -60... 1

(For references regarding interlacing roots, see A202605.)

Cf. A203990, A202605.

f[i_, j_] := (i + j) Min[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}]]  (* A203990 *)
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[%]               (* A203991 *)
TableForm[Table[c[n], {n, 1, 10}]]

A203997 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of min{i(j+1),j(i+1)} (A203996).

Original entry on oeis.org

2, -1, 3, -8, 1, 4, -19, 20, -1, 5, -34, 69, -40, 1, 6, -53, 160, -189, 70, -1, 7, -76, 305, -552, 434, -112, 1, 8, -103, 516, -1265, 1560, -882, 168, -1, 9, -134, 805, -2496, 4235, -3828, 1638, -240, 1, 10, -169, 1184, -4445, 9646

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:
2...-1
3...-8.....1
4...-19....20....-1
5...-34....69....-40....1
6...-53....160...-189...70....-1
7...-76....305...-552...434...-112...1

(For references regarding interlacing roots, see A202605.)

Cf. A203996, 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}]]   (* A203996 *)
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[%]      (* A203997 *)
TableForm[Table[c[n], {n, 1, 10}]]

cp's OEIS Frontend

A202870 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of the symmetric matrix A202869; by antidiagonals.

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A202875 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of the symmetric matrix A202874; by antidiagonals.

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A203004 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of the symmetric matrix A203003; by antidiagonals.

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Mathematica

A203906 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of A203905.

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A203946 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of A203945.

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A203954 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of A203953.

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A203956 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of A203955.

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A203989 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of {max(i,j)} (A051125).

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