A139584 A triangle of coefficients of A053122 type binomials {x,y},{y,z} and {z,x}, made using A_n Cartan type matrix characteristic polynomials: an(x,n) = CharacteristicPolynomial(M(A_n,n)); f(x,y,n) = Sum[Coefficients(an[x,n)*x^i*y^(n-i),{i,0,n}]; p(x,y,z,n) = f(x,y,n) + f(y,z,n) + f(z,x,n).

3, 5, -2, 6, -8, 2, 7, -20, 12, -2, 9, -40, 42, -16, 2, 12, -70, 112, -72, 20, -2, 15, -112, 252, -240, 110, -24, 2, 17, -168, 504, -660, 440, -156, 28, -2, 18, -240, 924, -1584, 1430, -728, 210, -32, 2, 19, -330, 1584, -3432, 4004, -2730, 1120, -272, 36, -2, 21, -440, 2574, -6864, 10010, -8736, 4760, -1632, 342

Offset

1,1

Comments

Row sums are: {3, 3, 0, -3, -3, 0, 3, 3, 0, -3, -3, ...}

Links

Table of n, a(n) for n=1..64.

Formula

an(x,n) = CharacteristicPolynomial(M(A_n,n));

f[x,y,n) = Sum[Coefficients(an[x,n)*x^i*y^(n-i),{i,0,n}];

p(x,y,z,n) = f(x,y,n) + f(y,z,n) + f(z,x,n);

Out_n,m = Coefficients(p(x,1,1,n).

Example

{3},
{5, -2},
{6, -8, 2},
{7, -20,12, -2},
{9, -40, 42, -16, 2},
{12, -70, 112, -72, 20, -2},
{15, -112, 252, -240, 110, -24, 2},
{17, -168, 504, -660, 440, -156, 28, -2},
{18, -240, 924, -1584, 1430, -728, 210, -32, 2},
{19, -330, 1584, -3432, 4004, -2730, 1120, -272,36, -2},
{21, -440, 2574, -6864, 10010, -8736, 4760, -1632, 342, -40, 2}

Mathematica

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}]; an[x_, n_] := If[n == 0, 1, CharacteristicPolynomial[M[n], x]]
f[x_, y_, n_] := Sum[CoefficientList[an[x, n], x][[i + 1]]*x^i*y^(n - i), {i, 0, Length[CoefficientList[an[x, n], x]] - 1}];
Table[ExpandAll[f[x, y, n] + f[y, z, n] + f[x, z, n]], {n, 0, 10}];
a = Table[CoefficientList[ExpandAll[f[x, y, n] + f[y, z, n] + f[x, z, n]] /. y -> 1 /. z -> 1, x], {n, 0, 10}]; Flatten[a]

Crossrefs

Cf. A053122.

Sequence in context: A302793 A010782 A333111 * A064790 A113966 A164611

Adjacent sequences: A139581 A139582 A139583 * A139585 A139586 A139587

Keyword

uned,sign

Author

Roger L. Bagula and Gary W. Adamson, Jun 11 2008

Status

approved