A136329 - OEIS

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

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OFFSET

1,2

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

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=CharacteristicPolynomial[{{2, -4}, {-1, 2}}, x] or CharacteristicPolynomial[{{2, -1}, {-4, 2}}, x]

EXAMPLE

{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}

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]

CROSSREFS

Cf. A053122, A136674, A110162.

Sequence in context: A143425 A323376 A166555 * A122073 A106236 A270640

Adjacent sequences: A136326 A136327 A136328 * A136330 A136331 A136332

KEYWORD

tabl,uned,sign

AUTHOR

Roger L. Bagula, Apr 12 2008

STATUS

approved