A136329 Triangular sequence of coefficients of a polynomial recursion for C_n and B_n Cartan matrices: p(x, n) = (-2 + x)*p(x, n - 1) - p(x, n - 2) p(x,n)=x2-4*x+4-m:m=4;(related sequence: A_n:m=1,G_n,m=3,B_n,C_n,m=2) This triangular sequence is an extension to the Cartan pattern of matrices.

1, -2, 1, 0, -4, 1, 2, 7, -6, 1, -4, -8, 18, -8, 1, 6, 5, -38, 33, -10, 1, -8, 4, 63, -96, 52, -12, 1, 10, -21, -84, 222, -190, 75, -14, 1, -12, 48, 84, -432, 550, -328, 102, -16, 1, 14, -87, -36, 726, -1342, 1131, -518, 133, -18, 1, -16, 140, -99, -1056, 2860, -3276, 2065, -768, 168, -20, 1

Offset: 1

Author

Roger L. Bagula, Apr 12 2008

Keywords

• tabl
• uned
• sign

Comments

Row sums are:

{1, -1, -3, 4, -1, -3, 4, -1, -3, 4, -1}

This sequence is also related to different p(x,2) start:

1) A_n like sequence A053122 ( sign change)

2) my G_n matrix A136674

3) B_n,C_n A110162

Examples

			{1},
{-2, 1},
{0, -4, 1},
{2, 7, -6, 1},
{-4, -8, 18, -8, 1},
{6, 5, -38, 33, -10,1},
{-8, 4, 63, -96, 52, -12, 1},
{10, -21, -84, 222, -190, 75, -14, 1},
{-12, 48, 84, -432, 550, -328, 102, -16, 1},
{14, -87, -36, 726, -1342, 1131, -518, 133, -18, 1},
{-16, 140, -99, -1056, 2860, -3276, 2065, -768, 168, -20, 1}
		

Crossrefs

Cf. A053122, A136674, A110162.

Programs

Mathematica

Clear[p, a] p[x, 0] = 1; p[x, 1] = -2 + x; p[x, 2] = x^2 - 4*x ; p[x_, n_] := p[x, n] = (-2 + x)*p[x, n - 1] - p[x, n - 2]; Table[ExpandAll[p[x, n]], {n, 0, 10}]; a = Table[CoefficientList[p[x, n], x], {n, 0, 10}] Flatten[a]

Formula

p(x, n) = (-2 + x)*p(x, n - 1) - p(x, n - 2) Three start vectors necessary: p(x,0)=1;p(x,1)=2-x; p(x,2)=x^2-4*x=CharacteristicPolynomial[{{2, -4}, {-1, 2}}, x] or CharacteristicPolynomial[{{2, -1}, {-4, 2}}, x]