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A167049
Number of reduced words of length n in Coxeter group on 19 generators S_i with relations (S_i)^2 = (S_i S_j)^13 = I.
2
1, 19, 342, 6156, 110808, 1994544, 35901792, 646232256, 11632180608, 209379250944, 3768826516992, 67838877305856, 1221099791505408, 21979796247097173, 395636332447746036, 7121453984059373415
OFFSET
0,2
COMMENTS
The initial terms coincide with those of A170738, although the two sequences are eventually different.
Computed with MAGMA using commands similar to those used to compute A154638.
LINKS
Index entries for linear recurrences with constant coefficients, signature (17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, -153).
FORMULA
G.f.: (t^13 + 2*t^12 + 2*t^11 + 2*t^10 + 2*t^9 + 2*t^8 + 2*t^7 + 2*t^6 + 2*t^5 + 2*t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/(153*t^13 - 17*t^12 - 17*t^11 - 17*t^10 - 17*t^9 - 17*t^8 - 17*t^7 - 17*t^6 - 17*t^5 - 17*t^4 - 17*t^3 - 17*t^2 - 17*t + 1).
G.f.: (1+x)*(1-x^13)/(1 - 18*x + 170*x^13 - 153*x^14). - G. C. Greubel, Apr 26 2019
a(n) = -153*a(n-13) + 17*Sum_{k=1..12} a(n-k). - Wesley Ivan Hurt, May 06 2021
MATHEMATICA
CoefficientList[Series[(1+x)*(1-x^13)/(1-18*x+170*x^13-153*x^14), {x, 0, 20}], x] (* G. C. Greubel, May 31 2016, modified Apr 26 2019 *)
coxG[{13, 153, -17}] (* The coxG program is at A169452 *) (* G. C. Greubel, Apr 26 2019 *)
PROG
(PARI) my(x='x+O('x^20)); Vec((1+x)*(1-x^13)/(1-18*x+170*x^13-153*x^14)) \\ G. C. Greubel, Apr 26 2019
(Magma) R<x>:=PowerSeriesRing(Integers(), 20); Coefficients(R!( (1+x)*(1-x^13)/(1-18*x+170*x^13-153*x^14) )); // G. C. Greubel, Apr 26 2019
(Sage) ((1+x)*(1-x^13)/(1-18*x+170*x^13-153*x^14)).series(x, 20).coefficients(x, sparse=False) # G. C. Greubel, Apr 26 2019
CROSSREFS
Sequence in context: A165881 A166413 A166600 * A167126 A167676 A167929
KEYWORD
nonn
AUTHOR
John Cannon and N. J. A. Sloane, Dec 03 2009
STATUS
approved