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 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. 0
 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 (list; table; graph; refs; listen; history; text; internal format)
 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 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

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Last modified January 20 17:42 EST 2022. Contains 350472 sequences. (Running on oeis4.)