A136321 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=5;(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, -1, -4, 1, 4, 6, -6, 1, -7, -4, 17, -8, 1, 10, -5, -32, 32, -10, 1, -13, 24, 42, -88, 51, -12, 1, 16, -56, -28, 186, -180, 74, -14, 1, -19, 104, -42, -312, 495, -316, 101, -16, 1, 22, -171, 216, 396, -1122, 1053, -504, 132, -18, 1, -25, 260, -561, -264, 2145, -2912, 1960, -752, 167, -20, 1

Offset

1,2

Comments

Row sums are:

{1, -1, -4, 5, -1, -4, 5, -1, -4, 5, -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

Links

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

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-1=CharacteristicPolynomial[{{2, -5}, {-1, 2}}, x] or CharacteristicPolynomial[{{2, -1}, {-5, 2}}, x]

Example

{1},
{-2, 1},
{-1, -4, 1},
{4, 6, -6, 1},
{-7, -4, 17, -8, 1},
{10, -5, -32, 32, -10, 1},
{-13, 24, 42, -88,51, -12, 1},
{16, -56, -28,186, -180, 74, -14, 1},
{-19, 104, -42, -312, 495, -316, 101, -16, 1},
{22, -171, 216, 396, -1122, 1053, -504, 132, -18, 1},
{-25, 260, -561, -264,2145, -2912, 1960, -752, 167, -20, 1}

Mathematica

Clear[p, a] p[x, 0] = 1; p[x, 1] = -2 + x; p[x, 2] = x^2 - 4*x - 1; 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]

Crossrefs

Cf. A053122, A136674, A110162.

Sequence in context: A281422 A344529 A144389 * A112987 A125138 A021477

Adjacent sequences: A136318 A136319 A136320 * A136322 A136323 A136324

Keyword

tabl,uned,sign

Author

Roger L. Bagula, Apr 12 2008

Status

approved