OFFSET
1,6
COMMENTS
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
LINKS
Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).
FORMULA
From Colin Barker, Dec 29 2015: (Start)
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)
MATHEMATICA
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 *)
PROG
(PARI) concat(vector(5), Vec(7*x^6/(1-x)^6 + O(x^50))) \\ Colin Barker, May 04 2016
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Philippe A.J.G. Chevalier, Dec 28 2015
STATUS
approved