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A266387
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Number of orbits of Aut(Z^7) as function of the infinity norm n of the representative lattice point of the orbit, when the cardinality of the orbit is equal to 322560.
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1
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0, 0, 0, 0, 0, 7, 42, 147, 392, 882, 1764, 3234, 5544, 9009, 14014, 21021, 30576, 43316, 59976, 81396, 108528, 142443, 184338, 235543, 297528, 371910, 460460, 565110, 687960, 831285, 997542, 1189377, 1409632, 1661352, 1947792, 2272424, 2638944, 3051279
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OFFSET
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1,6
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COMMENTS
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The sequence was discovered by enumerating all orbits of Aut(Z^7) and sorting the orbits as function of the infinity norm of the representative integer lattice points. This sequence is one of the 30 sequences that are obtained by classifying the orbits in a table with the rows being the infinity norm and the columns being the 30 cardinalities (1, 14, 84, 128, 168, 280, 448, 560, 672, 840, 896, 1680, 2240, 2688, 3360, 4480, 5376, 6720, 8960, 13440, 17920, 20160, 26880, 40320, 53760, 80640, 107520, 161280, 322560, 645120) generated by signed permutations of integer lattice points of Z^7.
The continued fraction expansion of this sequence is finite and simplifies to the g.f. 7*x^6/(1-x)^6 (see Mathematica). - Benedict W. J. Irwin, Feb 09 2016
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LINKS
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FORMULA
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a(n) = 7*(n-1)*(n-2)*(n-3)*(n-4)*(n-5)/120.
a(n) = 6*a(n-1)-15*a(n-2)+20*a(n-3)-15*a(n-4)+6*a(n-5)-a(n-6) for n>6.
G.f.: 7*x^6 / (1-x)^6.
(End)
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MATHEMATICA
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Join[{0, 0, 0, 0, 0}, Table[Abs[SeriesCoefficient[Series[7/(x+6/(x - 5/2/(x + ContinuedFractionK[If[Mod[k, 2] ==0, (7 + k/2)/(6 + 2 k), ((k + 1)/2 - 5)/(2 (k - 1) +6)], x, {k, 0, 8}]))), {x, Infinity, 101}], 2 n + 1]], {n, 0, 50}]] - (* Benedict W. J. Irwin, Feb 09 2016 *)
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PROG
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(PARI) concat(vector(5), Vec(7*x^6/(1-x)^6 + O(x^50))) \\ Colin Barker, May 04 2016
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CROSSREFS
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Other sequences that give the number of orbits of Aut(Z^7) as function of the infinity norm for different cardinalities of these orbits: A000579, A154286, A102860, A002412, A045943, A115067, A008586, A008585, A005843, A001477, A000217.
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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