A127410 -id:A127410 - cp's OEIS frontend

A127411 Negative value of coefficient of x^(n-6) in the characteristic polynomial of a certain n X n integer circulant matrix.

27216, 453789, 3866624, 22674816, 103500000, 393286542, 1297410048, 3822832728, 10267329072, 25518796875, 59378761728, 130535973152, 273106821312, 547049504268, 1054272000000, 1962916959024, 3543150344976, 6218839661001, 10640820731904, 17789062500000

Offset: 6

Paul Max Payton, Jan 14 2007

The n X n circulant matrix used here has first row 1 through n and each successive row is a circular rotation of the previous row to the right by one element.

The coefficient of x^(n-6) exists only for n>5, so the sequence starts with a(6). In order to obtain a nonnegative sequence the coefficient (which is negative for all n>5) is multiplied by -1.

			The circulant matrix for n = 6 is
[1 2 3 4 5 6]
[6 1 2 3 4 5]
[5 6 1 2 3 4]
[4 5 6 1 2 3]
[3 4 5 6 1 2]
[2 3 4 5 6 1]
The characteristic polynomial of this matrix is x^6 - 6*x^5 -196*x^4 - 1980*x^3 - 10044*x^2 - 25920*x - 27216. The coefficient of x^(n-6) is -27216, hence a(6) = 27216.

Daniel Zwillinger, ed., "CRC Standard Mathematical Tables and Formulae", 31st Edition, ISBN 1-58488-291, Section 2.6.2.25 (page 141) and Section 2.6.11.3 (page 152).

T. D. Noe, Table of n, a(n) for n = 6..1000

Cf. A000142 (factorial numbers), A014206 (n^2+n+2), A127407, A127408, A127409, A127410, A127412.

[ -Coefficient(CharacteristicPolynomial(Matrix(IntegerRing(), n, n, [< i, j, 1 + (j-i) mod n > : i, j in [1..n] ] )), n-6) : n in [6..22] ]; // Klaus Brockhaus, Jan 27 2007

[ (n-5)*(n-4)*(n-3)*(n-2)*(n-1)*n^6*(5*n+19) / (2*Factorial(7)) : n in [6..22] ]; // Klaus Brockhaus, Jan 27 2007

n * (n+1) * (n+2) * (n+3) * (n+4) * (n+5)^6 * (5*n + 44) / (2*factorial(7)); % Paul Max Payton, Jan 14 2007

a(n) = {-polcoef(charpoly(matrix(n,n,i,j,(j-i)%n+1),x),n-6)} \\ Klaus Brockhaus, Jan 27 2007

a(n) = {(5*n^12-56*n^11+140*n^10+490*n^9-2905*n^8+4606*n^7-2280*n^6)/(2*7!)} \\ Klaus Brockhaus, Jan 27 2007

a(n+5) = n*(n+1)*(n+2)*(n+3)*(n+4)*(n+5)^6*(5*n+44)/(2*7!) for n>=1.
a(n) = (5*n^12 - 56*n^11 + 140*n^10 + 490*n^9 - 2905*n^8 + 4606*n^7 - 2280*n^6)/(2*7!) for n>=6.
G.f.: x^6*(x^6 + 131*x^5 + 150*x^4 - 20470*x^3 - 90215*x^2 - 99981*x - 27216)/(x-1)^13. - Colin Barker, May 29 2012

Edited by Klaus Brockhaus, Jan 27 2007

A127407 Negative value of coefficient of x^(n-2) in the characteristic polynomial of a certain n X n integer circulant matrix.

Original entry on oeis.org

3, 15, 44, 100, 195, 343, 560, 864, 1275, 1815, 2508, 3380, 4459, 5775, 7360, 9248, 11475, 14079, 17100, 20580, 24563, 29095, 34224, 40000, 46475, 53703, 61740, 70644, 80475, 91295, 103168, 116160, 130339, 145775, 162540, 180708, 200355

Offset: 2

Paul Max Payton, Jan 14 2007

The n X n circulant matrix used here has first row 1 through n and each successive row is a circular rotation of the previous row to the right by one element.

The coefficient of x^(n-2) exists only for n>1, so the sequence starts with a(2). In order to obtain a nonnegative sequence the coefficient (which is negative for all n>1) is multiplied by -1.

			The circulant matrix for n = 5 is
[1 2 3 4 5]
[5 1 2 3 4]
[4 5 1 2 3]
[3 4 5 1 2]
[2 3 4 5 1]
The characteristic polynomial of this matrix is x^5 - 5*x^4 -100*x^3 - 625*x^2 - 1750*x - 1875. The coefficient of x^(n-2) is -100, hence a(5) = 100.

Daniel Zwillinger, ed., "CRC Standard Mathematical Tables and Formulae", 31st Edition, ISBN 1-58488-291, Section 2.6.2.25 (page 141) and Section 2.6.11.3 (page 152).

Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A000142 (factorial numbers), A014206 (n^2+n+2), A127408, A127409, A127410, A127411, A127412.

[ -Coefficient(CharacteristicPolynomial(Matrix(IntegerRing(), n, n, [< i, j, 1 + (j-i) mod n > : i, j in [1..n] ] )), n-2) : n in [2..38] ]; // Klaus Brockhaus, Jan 27 2007

[ (n-1) * n^2 * (n+7) / (2 * Factorial(3)) : n in [2..38] ]; // Klaus Brockhaus, Jan 27 2007

n * (n+1)^2 * (n+8) / (2 * factorial(3)); % Paul Max Payton, Jan 14 2007

a(n) = {-polcoeff(charpoly(matrix(n,n,i,j,(j-i)%n+1),x),n-2)} \\ Klaus Brockhaus, Jan 27 2007

a(n) = {(n^4+6*n^3-7*n^2)/(2*3!)} \\ Klaus Brockhaus, Jan 27 2007

a(n+1) = n*(n+1)^2*(n+8)/(2*3!) for n>=1.
a(n) = ((n-1)^4+10*(n-1)^3+17*(n-1)^2+8*(n-1))/(2*3!) for n>=2.
a(n) = (n^2*(-7+6*n+n^2))/12. G.f.: x^2*(3-x^2)/(1-x)^5. - Colin Barker, May 13 2012

Edited by Klaus Brockhaus, Jan 27 2007

A127408 Negative value of coefficient of x^(n-3) in the characteristic polynomial of a certain n X n integer circulant matrix.

Original entry on oeis.org

18, 144, 625, 1980, 5145, 11648, 23814, 45000, 79860, 134640, 217503, 338884, 511875, 752640, 1080860, 1520208, 2098854, 2850000, 3812445, 5031180, 6558013, 8452224, 10781250, 13621400, 17058600, 21189168, 26120619, 31972500, 38877255

Offset: 3

Paul Max Payton, Jan 14 2007

The n X n circulant matrix used here has first row 1 through n and each successive row is a circular rotation of the previous row to the right by one element.

The coefficient of x^(n-3) exists only for n>2, so the sequence starts with a(3). In order to obtain a nonnegative sequence the coefficient (which is negative for all n>2) is multiplied by -1.

			The circulant matrix for n = 5 is
[1 2 3 4 5]
[5 1 2 3 4]
[4 5 1 2 3]
[3 4 5 1 2]
[2 3 4 5 1]
The characteristic polynomial of this matrix is x^5 - 5*x^4 -100*x^3 - 625*x^2 - 1750*x - 1875. The coefficient of x^(n-3) is -625, hence a(5) = 625.

Daniel Zwillinger, Ed., "CRC Standard Mathematical Tables and Formulae", 31st Edition, ISBN 1-58488-291, Section 2.6.2.25 (page 141) and Section 2.6.11.3 (page 152).

Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A000142 (factorial numbers), A014206 (n^2+n+2), A127407, A127409, A127410, A127411, A127412.

[ -Coefficient(CharacteristicPolynomial(Matrix(IntegerRing(), n, n, [< i, j, 1 + (j-i) mod n > : i, j in [1..n] ] )), n-3) : n in [3..31] ] ; // Klaus Brockhaus, Jan 26 2007

[ (n-2) * (n-1) * n^3 * (2*(n-2) + 14) / (2 * Factorial(4)) : n in [3..31] ] ; // Klaus Brockhaus, Jan 26 2007

n * (n+1) * (n+2)^3 * (2*n + 14) / (2 * factorial(4)); % Paul Max Payton, Jan 14 2007

a(n) = {-polcoef(charpoly(matrix(n,n,i,j,(j-i)%n+1),x),n-3)} \\ Klaus Brockhaus, Jan 26 2007

a(n) = {(n^6+2*n^5-13*n^4+10*n^3)/4!} \\ Klaus Brockhaus, Jan 26 2007

a(n+2) = n*(n+1)*(n+2)^3*(2n+14)/(2 * 4!) for n>=1.
a(n) = (n^6+2*n^5-13*n^4+10*n^3)/4! for n>=3.
G.f.: x^3*(3-x)*(6+8*x+x^2)/(1-x)^7. - Colin Barker, May 13 2012

Edited and extended by Klaus Brockhaus, Jan 26 2007

A127409 Negative value of coefficient of x^(n-4) in the characteristic polynomial of a certain n X n integer circulant matrix.

Original entry on oeis.org

160, 1750, 10044, 40817, 132608, 367416, 903000, 2020458, 4191264, 8168446, 15107092, 26719875, 45473792, 74834816, 119567664, 186098388, 282948000, 421245846, 615331948, 883458037, 1248597504, 1739375000, 2391126920

Offset: 4

Paul Max Payton, Jan 14 2007

The n X n circulant matrix used here has first row 1 through n and each successive row is a circular rotation of the previous row to the right by one element.

The coefficient of x^(n-4) exists only for n>3, so the sequence starts with a(4). In order to obtain a nonnegative sequence the coefficient (which is negative for all n>3) is multiplied by -1.

			The circulant matrix for n = 5 is
[1 2 3 4 5]
[5 1 2 3 4]
[4 5 1 2 3]
[3 4 5 1 2]
[2 3 4 5 1]
The characteristic polynomial of this matrix is x^5 - 5*x^4 -100*x^3 - 625*x^2 - 1750*x - 1875. The coefficient of x^(n-4) is -1750, hence a(5) = 1750.

Daniel Zwillinger, ed., "CRC Standard Mathematical Tables and Formulae", 31st Edition, ISBN 1-58488-291, Section 2.6.2.25 (page 141) and Section 2.6.11.3 (page 152).

Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).

Cf. A000142 (factorial numbers), A014206 (n^2+n+2), A127407, A127408, A127410, A127411, A127412.

[ -Coefficient(CharacteristicPolynomial(Matrix(IntegerRing(), n, n, [< i, j, 1 + (j-i) mod n > : i, j in [1..n] ] )), n-4) : n in [4..26] ];  // Klaus Brockhaus, Jan 27 2007

[ (n-3)*(n-2)*(n-1)*n^4*(3*n+13) / (2 * Factorial(5)) : n in [4..26] ]; // Klaus Brockhaus, Jan 27 2007

n * (n+1) * (n+2) * (n+3)^4 * (3*n + 22) / (2 * factorial(5)); % Paul Max Payton, Jan 14 2007

a(n) = {-polcoeff(charpoly(matrix(n,n,i,j,(j-i)%n+1),x),n-4)} \\ Klaus Brockhaus, Jan 27 2007

a(n) = {(3*n^8 - 5*n^7 - 45*n^6 + 125*n^5 - 78*n^4)/(2*5!)} \\ Klaus Brockhaus, Jan 27 2007

a(n+3) = n*(n+1)*(n+2)*(n+3)^4*(3*n+22)/(2*5!) for n>=1.
a(n) = (3*n^8-5*n^7-45*n^6+125*n^5-78*n^4)/(2*5!) for n>=4.
G.f.: x^4*(160+310*x+54*x^2-19*x^3-x^4)/(1-x)^9. - Colin Barker, May 13 2012

Edited, corrected and extended by Klaus Brockhaus, Jan 27 2007

A127412 Triangular table containing values of coefficients of the characteristic polynomial of a certain n x n circulant matrix, read by rows.

Original entry on oeis.org

1, 1, -1, 1, -2, -3, 1, -3, -15, -18, 1, -4, -44, -144, -160, 1, -5, -100, -625, -1750, -1875, 1, -6, -195, -1980, -10044, -25920, -27216, 1, -7, -343, -5145, -40817, -184877, -453789, -470596, 1, -8, -560, -11648, -132608, -917504, -3866624, -9175040, -9437184, 1, -9, -864, -23814, -367416, -3582306

Offset: 0

Paul Max Payton, Feb 09 2007

This is a lower triangular table.

			The third row represents the coefficients of the characteristic polynomial of [1 2 3; 3 1 2; 2 3 1], which is x^3 - 3*x^2 - 15*x - 18. Thus the row reads 1,-3,-15,-18.

Daniel Zwillinger, ed., "CRC Standard Mathematical Tables and Formulae", 31st Edition, ISBN 1-58488-291, Section 2.6.2.25 (page 141) and Section 2.6.11.3 (page 152).

Cf. A000142, A014206, A127407, A127408, A127409, A127410, A127411.

First column is unity. Second column (A127407) is a(n+1) = n*(n+1)^2*(n+8)/(2*3!) for n>=1. Third column (A127408) is a(n+2) = n*(n+1)*(n+2)^3*(2n+14)/(2 * 4!) for n>=1. In general, k-th column is given by a(n+(k-1)) = n*(n+1)*(n+2)*...*(n+(k-1))^k*((k-1)n+S(k))/(2 * (k+1)!) for n>=1, where S(k) is the k-th term of A014206.

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A127411 Negative value of coefficient of x^(n-6) in the characteristic polynomial of a certain n X n integer circulant matrix.

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A127407 Negative value of coefficient of x^(n-2) in the characteristic polynomial of a certain n X n integer circulant matrix.

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A127408 Negative value of coefficient of x^(n-3) in the characteristic polynomial of a certain n X n integer circulant matrix.

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A127409 Negative value of coefficient of x^(n-4) in the characteristic polynomial of a certain n X n integer circulant matrix.

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A127412 Triangular table containing values of coefficients of the characteristic polynomial of a certain n x n circulant matrix, read by rows.

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