A136600 Triangle of coefficients of characteristic polynomials of a special type of Cartan matrix: E_n for E_6,E_7,E_8,E_11 example M(6)/ E_6: {{2, -1, 0, 0, 0, 0}, {-1, 2, -1, 0, 0, 0}, {0, -1, 2, -1, 0, -1}, {0, 0, -1, 2, -1, 0}, {0, 0, 0, -1, 2, 0}, {0, 0, -1, 0, 0, 2}},.

1, 2, -1, 4, -4, 1, 6, -11, 6, -1, 5, -20, 21, -8, 1, 4, -34, 56, -36, 10, -1, 3, -52, 125, -120, 55, -12, 1, 2, -73, 246, -329, 220, -78, 14, -1, 1, -96, 440, -784, 714, -364, 105, -16, 1, 0, -120, 730, -1679, 1992, -1364, 560, -136, 18, -1, -1, -144, 1140, -3304, 4949, -4356, 2379, -816, 171, -20, 1, -2, -167, 1694

Offset: 1

Author

Roger L. Bagula, Mar 24 2008

Keywords

• uned
• tabl
• sign

Comments

Row sums are: {1, 1, 1, 0, -1, -1, 0, 1, 1, 0, -1, -1, 0}.

Solution for a polynomial recursion gives for higher polynomials:

p1 = Join[{1}, Table[CharacteristicPolynomial[MO[n], x], {n, 1, 12}]];

Table[Solve[{p1[[n]] - (a0*x - b0)*p1[[n - 1]] - c0*p1[[n - 2]] == 0, p1[[n + 1]] - (a0*x - b0)* p1[[n]] - c0*p1[[n - 1]] == 0, p1[[n + 2]] - (a0*x - b0)*p1[[n + 1]] - c0*p1[[n]] == 0}, {a0, b0, c0}], {n, 3, 10}];

Polynomial recursion:

P[x, n] = (2 - x)*P[x, n - 1] + P[x, n - 2]

Examples

			{1},
{2, -1},
{4, -4, 1},
{6, -11, 6, -1},
{5, -20, 21, -8, 1},
{4, -34, 56, -36, 10, -1},
{3, -52, 125, -120,55, -12, 1},
{2, -73, 246, -329, 220, -78, 14, -1},
{1, -96, 440, -784, 714, -364, 105, -16, 1},
{0, -120, 730, -1679, 1992, -1364, 560, -136, 18, -1},
{-1, -144, 1140, -3304, 4949, -4356, 2379,-816, 171, -20, 1},
{-2, -167, 1694, -6069, 11210, -12297, 8554, -3875, 1140, -210, 22, -1},
{-3, -188, 2415, -10528, 23540, -31448, 27026, -15488, 5984, -1540, 253, -24, 1}
		

References

• R. N. Cahn, Semi-Simple Lie Algebras and Their Representations, Dover, NY, 2006, ISBN 0-486-44999-8.page 139
• E. B. Dynkin, Semisimple subalgebras of semisimple Lie algebras, Amer. Math. Soc. Transl, 1957
• Sigurdur Helgasson, Differential Geometry, Lie Groups and Symmetric Spaces, Graduate Studies in Mathematics, volume 34. A. M. S. :ISBN 0-8218-2848-7, 1978

Crossrefs

Cf. A129844.

Programs

Mathematica

a[n_] := 2; b[n_] := -1; c[n_] := -1; T[n_, m_, d_] := If[ n == m, a[n], If[n == m - 1 || n == m + 1 || n ==m - 3 || n == m + 3, If[n == m - 1 &&m < d, b[m - 1], If[n == m + 1 && n < d, b[n - 1], If[n == m - 3 || n == m + 3, If[n == m - 3 && m == d, c[m - 3], If[n == m + 3 && n == d, c[n - 3], 0]]]]]]] MO[d_] := Table[If[TrueQ[T[n, m, d] == Null], 0, T[n, m, d]], {n, 1, d}, {m, 1, d}]; a1 = Join[{{1}}, Table[CoefficientList[CharacteristicPolynomial[MO[n], x], x], {n, 1, 12}]]' Flatten[a1]

Formula

h(n,m)=If[ n == m, a[n], If[n == m - 1 ||n == m + 1 || n == m - 3 || n == m + 3, If[n == m - 1 && m < d,b[m - 1], If[n == m + 1 && n < d, b[n - 1], If[n ==m - 3 || n == m + 3, If[n == m - 3 && m == d, c[m - 3], If[n == m + 3 && n == d, c[n - 3], 0]]]]]]] ; for n,m<=d