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 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. 0
 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 (list; table; graph; refs; listen; history; text; internal format)
 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 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: A328649 A281422 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

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Last modified February 26 05:36 EST 2020. Contains 332276 sequences. (Running on oeis4.)