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 A167925 A triangular sequence of the Matrix Markov type based on the 2x2 matrix: m={{a,1},{-1,1}}; which has determinant equal to trace. 1
 0, 1, 1, 1, 2, 3, 0, 2, 6, 12, -1, 0, 9, 32, 75, -1, -4, 9, 80, 275, 684, 0, -8, 0, 192, 1000, 3240, 8232, 1, -8, -27, 448, 3625, 15336, 47677, 122368, 1, 0, -81, 1024, 13125, 72576, 276115, 835584, 2158569, 0, 16, -162, 2304, 47500, 343440, 1599066 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS Row sums are: {0, 2, 6, 20, 115, 1043, 12656, 189420, 3356913, 68661516, 1591360540,...} Each row is a specific Markov sequence with a different limiting ratio. Conjecture: a(6),a(7),a(8),a(9)=coefficient of x^2 in the Maclaurin expansion of 1/(n*x^2+n*x+1) for n=1,2,3,4; a(10),a(11),a(12),a(13),a(14)=-coefficient of x^3 in the Maclaurin expansion of 1/(n*x^2+n*x+1) for n=1,2,3,4,5; a(15),a(16),a(17),a(18),a(19),a(20)=coefficient of x^4 in the Maclaurin expansion of 1/(n*x^2+n*x+1) for n=1,2,3,4,5,6; a(21),a(22),a(23),a(24),a(25),a(26),a(27)=-coefficient of x^5 in the Maclaurin expansion of 1/(n*x^2+n*x+1) for n=1,2,3,4,5,6,7. Etc... [From Francesco Daddi, Aug 04 2011] LINKS EXAMPLE {0}, {1, 1}, {1, 2, 3}, {0, 2, 6, 12}, {-1, 0, 9, 32, 75}, {-1, -4, 9, 80, 275, 684}, {0, -8, 0, 192, 1000, 3240, 8232}, {1, -8, -27, 448, 3625, 15336, 47677, 122368}, {1, 0, -81, 1024, 13125, 72576, 276115, 835584, 2158569}, {0, 16, -162, 2304, 47500, 343440, 1599066, 5705728, 16953624, 44010000}, {-1, 32, -243, 5120, 171875, 1625184, 9260657, 38961152, 133155495, 390500000, 1017681269} MATHEMATICA Clear[m, a, n, v]; m = {{a, 1}, {-1, 1}}; v[0] := {0, 1}; v[n_] := v[n] = m.v[n - 1]; Table[v[n][[1]], {n, 0, 10}, {a, 0, n}]; Flatten[%] CROSSREFS Sequence in context: A248342 A002392 A002708 * A209927 A059283 A160202 Adjacent sequences:  A167922 A167923 A167924 * A167926 A167927 A167928 KEYWORD sign,tabl,uned AUTHOR Roger L. Bagula, Nov 15 2009 STATUS approved

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