A130605 Polynomial Generalized Cartan Matrices like the B_n type: m4by4={{2, -1, 0, 0}, {-1, 2, -1, 0}, {0, -1, 2, -n}, {0, 0, -1, 2}} For B_4 the n=2: here n=3 is used in the general n X n matrix polynomials P(n,x). Since the C_n types are symmetrical, they have the same polynomials. For 2 X 2 this n=3 is the G_2 14 dimensional exceptional group.

-3, 1, 1, 1, -4, 1, 0, -8, 6, -1, -1, -12, 19, -8, 1, -2, -15, 44, -34, 10, -1, -3, -16, 84, -104, 53, -12, 1, 6, -59, 202, -295, 210, -77, 14, -1, 12, -124, 463, -792, 715, -364, 105, -16, 1, 24, -260, 1050, -2047, 2222, -1443, 574, -137, 18, -1, 48, -544, 2360, -5144, 6491, -5108, 2591, -848, 173, -20, 1

Offset

1,1

Comments

Absolute value row sum: Table[Apply[Plus, Abs[CoefficientList[a[[m]], x]]], {m, 1, Length[a]}] /. n -> 3 {3, 2, 6, 15, 41, 106, 273, 864, 2592, 7776, 23328} Which suggests the pattern of the determinants as detM= even-f(m)*n might not hold for the 0th matrix being -n: -1 may fit the pattern better. To get the polynomials I programmed the matrices individually in Mathematica to the m=10 level and took their characteristic polynomials ( long listing).

Links

Table of n, a(n) for n=1..66.

Formula

P(n,x)->listing below in Mathematica. a(n,m) = CoefficientList[P(n,x),x]

Example

{-3},
{1, 1},
{1, -4, 1},
{0, -8, 6, -1},
{-1, -12, 19, -8, 1},
{-2, -15, 44, -34, 10, -1},
{-3, -16, 84, -104, 53, -12, 1},
{6, -59, 202, -295, 210, -77, 14, -1},
{12, -124,463, -792, 715, -364, 105, -16, 1},
{24, -260, 1050, -2047, 2222, -1443, 574, -137, 18, -1},
{48, -544, 2360, -5144, 6491, -5108, 2591, -848, 173, -20, 1}

Mathematica

a = Factor[ {-n, x -(2 - n), 4 - n - 4 x + x^2, 6 - 2 n - 11 x + n x + 6 x^2 - x^3, 8 - 3 n - 24 x + 4 n x + 22 x^2 - n x^2 - 8 x^3 + x^4, 10 - 4 n - 45 x + 10 n x + 62 x^2 - 6 n x^2 - 37 x^3 + n x^3 + 10 x^4 - x^5, 12 - 5 n - 76 x + 20 n x + 147 x^2 - 21 n x^2 - 128 x^3 + 8 n x^3 + 56 x^4 - n x^4 - 12 x^5 + x^6, 24 - 6 n - 164 x + 35 n x + 370 x^2 - 56 n x^2 - 403 x^3 + 36 n x^3 + 240 x^4 - 10 n x^4 - 80 x^5 + n x^5 + 14 x^6 - x^7, 48 - 12 n - 352 x + 76 n x + 904 x^2 - 147 n x^2 - 1176 x^3 + 128 n x^3 + 883x^4 - 56 n x^4 - 400 x^5 + 12 n x^5 + 108 x^6 - n x^6 - 16 x^7 + x^8, 96 - 24 n - 752 x + 164 n x + 2160 x^2 - 370 n x^2 - 3256 x\^3 + 403 n x^3 + 2942 x^4 - 240 n x^4 - 1683 x^5 + 80 n x^5 + 616 x^6 - 14 n x^6 - 140 x^7 + n x^7 + 18 x^8 - x^9, 192 - 48 n - 1600 x + 352 n x + 5072 x^2 - 904 n x^2 - 8672 x^3 + 1176 n x^3 + 9140 x^4 - 883 n x^4 - 6308 x^5 + 400 n x^5 + 2915 x^6 - 108 n x^6 - 896 x^7 + 16 n x^7 + 176 x^8 - n x^8 - 20 x^9 + x^10} ] b = Table[ CoefficientList[ a[ \([ \)\(m\)\( ]\) ], x ], {m, 1, Length[ a ]} ]; b /. n -> 3; Flatten[ % ]

Crossrefs

Sequence in context: A195644 A143632 A336455 * A354872 A368335 A157261

Adjacent sequences: A130602 A130603 A130604 * A130606 A130607 A130608

Keyword

uned,sign

Author

Roger L. Bagula, Jun 17 2007

Status

approved