A202605 -id:A202605 - cp's OEIS frontend

A203999 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of max{i(j+1-1),j(i+1)-1} (A203998).

1, -1, -4, -6, 1, 7, 27, 17, -1, -10, -60, -99, -36, 1, 13, 105, 279, 269, 65, -1, -16, -162, -593, -944, -609, -106, 1, 19, 231, 1077, 2405, 2610, 1218, 161, -1, -22, -312, -1767, -5092, -7865, -6264, -2226, -232, 1, 25, 405, 2699, 9541

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
-4....-6.....1
7.... 27....17...-1
-10...-60...-99...-36...1

(For references regarding interlacing roots, see A202605.)

Cf. A203998, A202605.

f[i_, j_] := Max[i (j + 1) - 1, j (i + 1) - 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}]]    (* A203998 *)
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[%]   (* A203999 *)
TableForm[Table[c[n], {n, 1, 10}]]

A204001 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of min{i(j+1-1),j(i+1)-1} (A204000).

Original entry on oeis.org

1, -1, 1, -6, 1, 1, -9, 17, -1, 1, -12, 39, -36, 1, 1, -15, 69, -119, 65, -1, 1, -18, 107, -272, 294, -106, 1, 1, -21, 153, -515, 846, -630, 161, -1, 1, -24, 207, -868, 1925, -2232, 1218, -232, 1, 1, -27, 269, -1351, 3783, -6017, 5214

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...-6....1
1...-9....17...-1
1...-12...39...-36...1

(For references regarding interlacing roots, see A202605.)

Cf. A204000, A202605.

f[i_, j_] := Min[i (j + 1) - 1, j (i + 1) - 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}]]  (* A204000 *)
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[%]               (* A204001 *)
TableForm[Table[c[n], {n, 1, 10}]]

A204003 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of min{2i+j,i+2j} (A204002).

Original entry on oeis.org

3, -1, 2, -9, 1, 1, -9, 18, -1, 0, -5, 25, -30, 1, -1, 3, 14, -55, 45, -1, -2, 15, -27, -28, 105, -63, 1, -3, 31, -110, 135, 42, -182, 84, -1, -4, 51, -247, 550, -495, -42, 294, -108, 1, -5, 75, -450, 1365, -2145, 1485, 0, -450, 135, -1, -6, 103

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:
3...-1
2...-9.....1
1...-9....18...-1
0...-5....25...-30...1

(For references regarding interlacing roots, see A202605.)

Cf. A204002, A202605.

f[i_, j_] := Min[2 i + j, 2 j + i];
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}]]    (* A204002 *)
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[%]                 (* A204003 *)
TableForm[Table[c[n], {n, 1, 10}]]

A204005 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of max{2i+j-2,2j+i-2} (A204004).

Original entry on oeis.org

1, -1, -5, -5, 1, 9, 31, 12, -1, -13, -73, -105, -22, 1, 17, 131, 322, 265, 35, -1, -21, -205, -711, -1036, -560, -51, 1, 25, 295, 1320, 2775, 2730, 1050, 70, -1, -29, -401, -2197, -6050, -8745, -6258, -1806, -92, 1, 33, 523, 3390, 11557

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
-5....-5....1
9.....31...12....-1
-13...-73..-105...-22...1

(For references regarding interlacing roots, see A202605.)

Cf. A204004, A202605.

f[i_, j_] := Max[2 i + j - 2, 2 j + i - 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, 12}, {i, 1, n}]]  (* A204004 *)
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[%]   (* A204005 *)
TableForm[Table[c[n], {n, 1, 10}]]

A204007 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of min{2i+j-2,2j+i-2} (A204006).

Original entry on oeis.org

1, -1, 0, -5, 1, -1, -1, 12, -1, -2, 7, 5, -22, 1, -3, 19, -28, -15, 35, -1, -4, 35, -99, 84, 35, -51, 1, -5, 55, -220, 375, -210, -70, 70, -1, -6, 79, -403, 990, -1155, 462, 126, -92, 1, -7, 107, -660, 2093, -3575, 3069, -924, -210, 117, -1

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
 0....-5....1
-1....-1....12....-1
-2.....7....5.....-22...1

(For references regarding interlacing roots, see A202605.)

Cf. A204006, A202605.

f[i_, j_] := Min[2 i + j - 2, 2 j + i - 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, 12}, {i, 1, n}]]   (* A204006 *)
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[%]                (* A204007 *)
TableForm[Table[c[n], {n, 1, 10}]]

A204011 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of max{3i+j-3,i+3j-3} (A204008).

Original entry on oeis.org

1, -1, -11, -6, 1, 40, 70, 15, -1, -116, -328, -240, -28, 1, 304, 1176, 1456, 610, 45, -1, -752, -3680, -6408, -4704, -1295, -66, 1, 1792, 10592, 23760, 25080, 12432, 2436, 91, -1, -4160, -28800, -79040

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
-11....-6.....1
 40.....70....15....-1
-116...-328..-240....1

(For references regarding interlacing roots, see A202605.)

Cf. A204008, A202605.

f[i_, j_] := Max[3 i + j - 3, 3 j + i - 3];
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}]]   (* A204008 *)
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[%]                (* A204011 *)
TableForm[Table[c[n], {n, 1, 10}]]

A204013 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of min{3i+j-3,i+3j-3} (A204012).

Original entry on oeis.org

1, -1, 1, -6, 1, 0, -10, 15, -1, -4, -8, 40, -28, 1, -16, 24, 56, -110, 45, -1, -48, 160, -72, -224, 245, -66, 1, -128, 608, -880, 120, 672, -476, 91, -1, -320, 1920, -4160, 3520, 0, -1680, 840, -120, 1, -768, 5504, -15360, 20384

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 A202605 for a guide to related sequences.

			Top of the array:
 1....-1
 1....-6....1
 0....-10...15....-1
-4....-8....40....-28....1

(For references regarding interlacing roots, see A202605.)

Cf. A204012, A202605.

f[i_, j_] := Min[3 i + j - 3, 3 j + i - 3];
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}]]   (* A204012 *)
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[%]  (* A204013 *)
TableForm[Table[c[n], {n, 1, 10}]]

A204015 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of min{1+(j mod i), 1+( i mod j)} (A204014).

Original entry on oeis.org

1, -1, 0, -2, 1, -1, 3, 3, -1, 0, 2, -6, -4, 1, 0, -8, 8, 20, 5, -1, -16, 14, 58, -4, -31, -6, 1, 48, 16, -169, -121, 69, 63, 7, -1, 208, -320, -576, 540, 432, -128, -97, -8, 1, 400, -2048, 1876, 2340, -1828, -928, 309, 153, 9, -1, -4800, 6880

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 A202605 for a guide to related sequences.

			Top of the array:
 1...-1
 0...-2...1
-1....3...3...-1
 0....2..-6...-4....1
 0...-8...8....20...5...1

(For references regarding interlacing roots, see A202605.)

Cf. A204014, A202605.

f[i_, j_] := Min[1 + Mod[i, j], 1 + Mod[j, i]];
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}]]  (* A204014 *)
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[%]   (* A204015 *)
TableForm[Table[c[n], {n, 1, 10}]]

A204019 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of max{1+j mod i, 1+i mod j} (A204018).

Original entry on oeis.org

1, -1, -3, -2, 1, 8, 14, 3, -1, -21, -64, -40, -4, 1, 40, 266, 280, 90, 5, -1, 125, -930, -1671, -896, -175, -6, 1, -2940, 854, 8600, 7228, 2352, 308, 7, -1, 35035, 37744, -27334, -50164, -24594, -5376, -504, -8, 1, -372400

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).  The least zero of p(n) is -n.
For n>1, the least zero of p(n) is exactly 1-n; the greatest, for p(1) to p(5) is represented by (1,3,5.701...,9.158...13.392...).
See A202605 and A204016 for guides to related sequences.

			Top of the array:
 1....-1
-3....-2......1
 8.....14.....3....-1
-21...-64....-40...-4...1

(For references regarding interlacing roots, see A202605.)

Cf. A204018, A202605, A204016.

f[i_, j_] := 1 + Max[Mod[i, j], Mod[j, i]];
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}]]   (* A204018 *)
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[%]                  (* A204019 *)
TableForm[Table[c[n], {n, 1, 10}]]

A204023 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of max(2i-1, 2j-1) (A204022).

Original entry on oeis.org

1, -1, -6, -4, 1, 20, 36, 9, -1, -56, -160, -120, -16, 1, 144, 560, 700, 300, 25, -1, -352, -1728, -3024, -2240, -630, -36, 1, 832, 4928, 11088, 11760, 5880, 1176, 49, -1, -1920, -13312, -36608, -50688, -36960

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
-6....-4.....1
 20....36....9.....-1
-56...-160..-120...-16....1

(For references regarding interlacing roots, see A202605.)

Cf. A204022, A202605, A204016.

f[i_, j_] := Max[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}]]  (* A204022 *)
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[%]                 (* A204023 *)
TableForm[Table[c[n], {n, 1, 10}]]

cp's OEIS Frontend

A203999 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of max{i(j+1-1),j(i+1)-1} (A203998).

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A204001 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of min{i(j+1-1),j(i+1)-1} (A204000).

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A204003 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of min{2i+j,i+2j} (A204002).

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A204005 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of max{2i+j-2,2j+i-2} (A204004).

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A204007 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of min{2i+j-2,2j+i-2} (A204006).

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Mathematica

A204011 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of max{3i+j-3,i+3j-3} (A204008).

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Mathematica

A204013 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of min{3i+j-3,i+3j-3} (A204012).

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Mathematica

A204015 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of min{1+(j mod i), 1+( i mod j)} (A204014).

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