A158199 A coefficients of characteristic polynomials of A_n Cartan matrices times their transposes: t(n,m,d)=If[ n == m, 2, If[n == m - 1 || n == m + 1, -1, 0]]. M(d)=t(n,m,d)*Transpose[t(n,m,d)]

1, 4, -1, 9, -10, 1, 16, -52, 16, -1, 25, -190, 131, -22, 1, 36, -553, 736, -246, 28, -1, 49, -1372, 3206, -1870, 397, -34, 1, 64, -3024, 11584, -10900, 3808, -584, 40, -1, 81, -6084, 36234, -52074, 27847, -6766, 807, -46, 1, 100, -11385, 101024, -212738

Offset

0,2

Comments

Row sums are:

{1, 3, 0, -21, -55, 0, 377, 987, 0, -6765, -17711,...},

Example matrix:

M(3)={{5, -4, 1},

{-4, 6, -4},

{1, -4, 5}}

Links

Table of n, a(n) for n=0..48.

Formula

t(n,m,d)=If[ n == m, 2, If[n == m - 1 || n == m + 1, -1, 0]];

M(d)=t(n,m,d)*Transpose[t(n,m,d)];

out_(n,m)=coefficients(characteristicpolynomial(M(n),x),x)

Example

{1},
{4, -1},
{9, -10, 1},
{16, -52, 16, -1},
{25, -190, 131, -22, 1},
{36, -553, 736, -246, 28, -1},
{49, -1372, 3206, -1870, 397, -34, 1},
{64, -3024, 11584, -10900, 3808, -584, 40, -1},
{81, -6084, 36234, -52074, 27847, -6766, 807, -46, 1},
{100, -11385, 101024, -212738, 165844, -59555, 10960, -1066, 52, -1},
{121, -20086, 256399, -765314, 838915, -425272, 112828, -16606, 1361, -58, 1}

Mathematica

Clear[M, T, d, a, x, a0];
T[n_, m_, d_] := If[ n == m, 2, If[n == m - 1 || n == m + 1, -1, 0]];
M[d_] := Table[T[n, m, d], {n, 1, d}, {m, 1, d}].Transpose[Table[T[n, m, d], {n, 1, d}, {m, 1, d}]];
a0 = Table[M[d], {d, 1, 10}];
Table[Det[M[d]], {d, 1, 10}];
Table[CharacteristicPolynomial[M[d], x], {d, 1, 10}];
a = Join[{{1}}, Table[CoefficientList[Expand[CharacteristicPolynomial[M[ n], x]], x], {n, 1, 10}]];
Flatten[a]
Join[{1}, Table[Apply[Plus, CoefficientList[ Expand[CharacteristicPolynomial[M[n], x]], x]], {n, 1, 10}]];

Crossrefs

Sequence in context: A367025 A067015 A179193 * A091885 A069606 A344109

Adjacent sequences: A158196 A158197 A158198 * A158200 A158201 A158202

Keyword

sign,tabl,uned

Author

Roger L. Bagula, Mar 13 2009

Status

approved