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

1,2

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]

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

Links

Table of n, a(n) for n=1..69.

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

Example

{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}

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]

Crossrefs

Cf. A129844.

Sequence in context: A105542 A208907 A200057 * A136672 A097750 A304623

Adjacent sequences: A136597 A136598 A136599 * A136601 A136602 A136603

Keyword

uned,tabl,sign

Author

Roger L. Bagula, Mar 24 2008

Status

approved