A202605 -id:A202605 - cp's OEIS frontend

A204126 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of f(i,j)=(i if i=j and 1 otherwise) (A204125).

1, -1, 1, -3, 1, 2, -8, 6, -1, 6, -28, 29, -10, 1, 24, -124, 155, -75, 15, -1, 120, -668, 949, -565, 160, -21, 1, 720, -4248, 6636, -4564, 1610, -301, 28, -1, 5040, -31176, 52464, -40208, 16569, -3892, 518, -36, 1, 40320, -259488, 463956

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
6....-28....29...-10...1

(For references regarding interlacing roots, see A202605.)

Cf. A204125, A202605, A204016.

f[i_, j_] := 1; f[i_, i_] := 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, 15}, {i, 1, n}]]  (* A204125 *)
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[%]                 (* A204126 *)
TableForm[Table[c[n], {n, 1, 10}]]

A204128 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of f(i,j)=(i if i=j and 1 otherwise) (A204125).

Original entry on oeis.org

1, -1, 1, -3, 1, 2, -8, 6, -1, 8, -36, 35, -11, 1, 56, -268, 295, -119, 19, -1, 672, -3328, 3914, -1786, 361, -32, 1, 13440, -67904, 82936, -40496, 9237, -1027, 53, -1, 443520, -2267712, 2832024, -1437872, 350799, -43879, 2822

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
8....-36....35...-11...1

(For references regarding interlacing roots, see A202605.)

Cf. A204127, A202605, A204016.

f[i_, j_] := 1; f[i_, i_] := Fibonacci[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}]]  (* A204127 *)
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[%]                 (* A204128 *)
TableForm[Table[c[n], {n, 1, 10}]]

A204130 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of f(i,j)=(L(i) if i=j and 1 otherwise) (A204129).

Original entry on oeis.org

1, -1, 2, -4, 1, 6, -16, 8, -1, 36, -108, 69, -15, 1, 360, -1152, 834, -230, 26, -1, 6120, -20304, 15726, -4890, 693, -44, 1, 171360, -580752, 467724, -155524, 24797, -1963, 73, -1, 7882560, -27057312, 22300752, -7709504

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
2....-4.....1
6....-16....8....-1
36...-108...69...-15...1

(For references regarding interlacing roots, see A202605.)

Cf. A204129, A202605, A204016.

f[i_, j_] := 1; f[i_, i_] := LucasL[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, 15}, {i, 1, n}]]  (* A204129 *)
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[%]                 (* A204130 *)
TableForm[Table[c[n], {n, 1, 10}]]

A204132 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of f(i,j)=(2i-1 if i=j and 1 otherwise) for i>=1 and j>=1 (as in A204131).

Original entry on oeis.org

1, -1, 2, -4, 1, 8, -20, 9, -1, 48, -136, 80, -16, 1, 384, -1184, 820, -220, 25, -1, 3840, -12608, 9784, -3160, 490, -36, 1, 46080, -158976, 134400, -49504, 9380, -952, 49, -1, 645120, -2317824, 2097024, -853440, 186704

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
2....-4.....1
8....-20....9...-1
48...-136...80..-16...1

(For references regarding interlacing roots, see A202605.)

Cf. A204131, A202605, A204016.

f[i_, j_] := 1; f[i_, i_] := 2*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}]]  (* A204131 *)
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[%]                 (* A204132 *)
TableForm[Table[c[n], {n, 1, 10}]]

A204134 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of f(i,j)=(2i-1 if i=j and 1 otherwise) for i>=1 and j>=1 (as in A204131).

Original entry on oeis.org

1, -1, 1, -3, 1, 3, -11, 7, -1, 21, -83, 64, -15, 1, 315, -1287, 1074, -300, 31, -1, 9765, -40527, 35067, -10570, 1287, -63, 1, 615195, -2572731, 2265129, -707539, 92653, -5313, 127, -1, 78129765, -327967227, 291222882, -92551369

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
3....-11....7....-1
21...-83....64...-15...1

(For references regarding interlacing roots, see A202605.)

Cf. A204133, A202605, A204016.

f[i_, j_] := 1; f[i_, i_] := 2^(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}]]  (* A204133 *)
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[%]                 (* A204134 *)
TableForm[Table[c[n], {n, 1, 10}]]

A204144 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of f(i,j)=max(ceiling(i/j),ceiling(j/i)) (as in A204143).

Original entry on oeis.org

1, -1, -3, -2, 1, 8, 14, 3, -1, -12, -42, -35, -4, 1, 19, 95, 145, 73, 5, -1, -20, -140, -338, -336, -125, -6, 1, 16, 184, 665, 1037, 735, 205, 7, -1, -16, -212, -981, -2140, -2381, -1320, -303, -8, 1, 12, 200, 1209, 3581, 5727, 5021

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
-3...-2....1
 8....14...3....-1
-12..-42..-35...-4....1

(For references regarding interlacing roots, see A202605.)

Cf. A204143, A202605, A204016.

f[i_, j_] := Max[Ceiling[i/j], Ceiling[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, 15}, {i, 1, n}]]  (* A204143 *)
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[%]                 (* A204144 *)
TableForm[Table[c[n], {n, 1, 10}]]

A204155 Array read by rows: row n lists the coefficients of the characteristic polynomial of the n-th principal submatrix of max(2i-j, 2j-i), as in A204154.

Original entry on oeis.org

1, -1, -7, -3, 1, 33, 39, 6, -1, -135, -255, -125, -10, 1, 513, 1323, 1092, 305, 15, -1, -1863, -6075, -7047, -3444, -630, -21, 1, 6561, 25839, 38610, 27135, 8946, 1162, 28, -1, -22599, -104247, -190593, -175230

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;
    -7,   -3,    1;
    33,   39,    6,   -1;
  -135, -255, -125,  -10,    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. A204154, A202605, A204016.

f:= proc(n) local P,lambda,i;
P:= (-1)^n*LinearAlgebra:-CharacteristicPolynomial(Matrix(n,n,(i,j) -> max(2*i-j,2*j-i)),lambda);
seq(coeff(P,lambda,i),i=0..n);
end proc:
map(f, [$1..20]); # Robert Israel, Dec 03 2017

f[i_, j_] := Max[2 i - j, 2 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, 15}, {i, 1, n}]]  (* A204154 *)
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[%]                 (* A204155 *)
TableForm[Table[c[n], {n, 1, 10}]]

A204157 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of max(3i-j, 3j-i), as in A204156.

Original entry on oeis.org

1, -1, -13, -4, 1, 88, 78, 9, -1, -496, -704, -260, -16, 1, 2560, 4960, 3080, 650, 25, -1, -12544, -30720, -26784, -9856, -1365, -36, 1, 59392, 175616, 197120, 104160, 25872, 2548, 49, -1, -274432, -950272, -1304576, -901120, -327360, -59136, -4368, -64, 1, 1245184, 4939776, 8017920, 6849024, 3294720

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
-13...-4.....1
 88....78....9.....-1
-496..-704..-260...-16...1

(For references regarding interlacing roots, see A202605.)

Cf. A204156, A202605, A204016.

f[i_, j_] := -1 + Max[3 i - j, 3 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, 15}, {i, 1, n}]]    (* A204156 *)
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[%]                   (* A204157 *)
TableForm[Table[c[n], {n, 1, 10}]]

A204159 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of max(3i-2j, 3j-2i), as in A204158.

Original entry on oeis.org

1, -1, -14, -3, 1, 115, 79, 6, -1, -800, -895, -255, -10, 1, 5125, 7875, 3850, 625, 15, -1, -31250, -60875, -42075, -12180, -1295, -21, 1, 184375, 434375, 387750, 162375, 31710, 2394, 28, -1, -1062500, -2934375

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
-14....-3......1
 115....79.....6.....-1
-800...-895...-255...-10....1

(For references regarding interlacing roots, see A202605.)

Cf. A204158, A202605, A204016.

f[i_, j_] := Max[3 i - 2 j, 3 j - 2 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, 15}, {i, 1, n}]]   (* A204158 *)
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[%]                 (* A204159 *)
TableForm[Table[c[n], {n, 1, 10}]]

A204161 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of (3i-2 if i=j and = 0 otherwise), as in A204160.

Original entry on oeis.org

1, -1, 3, -5, 1, 18, -36, 12, -1, 162, -360, 153, -22, 1, 1944, -4644, 2295, -435, 35, -1, 29160, -73548, 40419, -9135, 990, -51, 1, 524880, -1382184, 823284, -210924, 27720, -1953, 70, -1, 11022480

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
3.....-5.....1
18....-36....12....-1
162...-360...153...-22...1

(For references regarding interlacing roots, see A202605.)

Cf. A204160, A202605, A204016.

f[i_, j_] := 1; f[i_, i_] := 2 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}]]    (* A204160 *)
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[%]                   (* A204161 *)
TableForm[Table[c[n], {n, 1, 10}]]

cp's OEIS Frontend

A204126 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of f(i,j)=(i if i=j and 1 otherwise) (A204125).

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A204128 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of f(i,j)=(i if i=j and 1 otherwise) (A204125).

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Mathematica

A204130 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of f(i,j)=(L(i) if i=j and 1 otherwise) (A204129).

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A204132 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of f(i,j)=(2i-1 if i=j and 1 otherwise) for i>=1 and j>=1 (as in A204131).

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A204134 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of f(i,j)=(2i-1 if i=j and 1 otherwise) for i>=1 and j>=1 (as in A204131).

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Mathematica

A204144 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of f(i,j)=max(ceiling(i/j),ceiling(j/i)) (as in A204143).

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Mathematica

A204155 Array read by rows: row n lists the coefficients of the characteristic polynomial of the n-th principal submatrix of max(2i-j, 2j-i), as in A204154.

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Maple

Mathematica

A204157 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of max(3i-j, 3j-i), as in A204156.

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