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A169403 Number of reduced words of length n in Coxeter group on 6 generators S_i with relations (S_i)^2 = (S_i S_j)^32 = I.

%I A169403 #18 Dec 27 2024 14:36:01
%S A169403 1,6,30,150,750,3750,18750,93750,468750,2343750,11718750,58593750,
%T A169403 292968750,1464843750,7324218750,36621093750,183105468750,
%U A169403 915527343750,4577636718750,22888183593750,114440917968750,572204589843750
%N A169403 Number of reduced words of length n in Coxeter group on 6 generators S_i with relations (S_i)^2 = (S_i S_j)^32 = I.
%C A169403 The initial terms coincide with those of A003948, although the two sequences are eventually different.
%C A169403 First disagreement is at index 32, the difference is 15. - _Klaus Brockhaus_, Jun 26 2011
%C A169403 Computed with Magma using commands similar to those used to compute A154638.
%H A169403 Vincenzo Librandi, <a href="/A169403/b169403.txt">Table of n, a(n) for n = 0..165</a>
%H A169403 <a href="/index/Rec#order_32">Index entries for linear recurrences with constant coefficients</a>, signature (4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, -10).
%F A169403 G.f.: (t^32 + 2*t^31 + 2*t^30 + 2*t^29 + 2*t^28 + 2*t^27 + 2*t^26 + 2*t^25 + 2*t^24 + 2*t^23 + 2*t^22 + 2*t^21 + 2*t^20 + 2*t^19 + 2*t^18 + 2*t^17 + 2*t^16 + 2*t^15 + 2*t^14 + 2*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)/(10*t^32 - 4*t^31 - 4*t^30 - 4*t^29 - 4*t^28 - 4*t^27 - 4*t^26 - 4*t^25 - 4*t^24 - 4*t^23 - 4*t^22 - 4*t^21 - 4*t^20 - 4*t^19 - 4*t^18 - 4*t^17 - 4*t^16 - 4*t^15 - 4*t^14 - 4*t^13 - 4*t^12 - 4*t^11 - 4*t^10 - 4*t^9 - 4*t^8 - 4*t^7 - 4*t^6 - 4*t^5 - 4*t^4 - 4*t^3 - 4*t^2 - 4*t + 1).
%F A169403 G.f.: (1+2*sum(k=1..31,x^k)+x^32)/(1-4*sum(k=1..31,x^k)+10*x^32).
%t A169403 coxG[{32,10,-4,30}] (* The coxG program is at A169452 *) (* _Harvey P. Dale_, Dec 27 2024 *)
%o A169403 (PARI) x='x+O('x^66); /* that many terms */
%o A169403 Vec((1+2*sum(k=1,31,x^k)+x^32)/(1-4*sum(k=1,31,x^k)+10*x^32)) /* show terms */
%o A169403 /* _Joerg Arndt_, Jun 26 2011 */
%Y A169403 Cf. A003948 (G.f.: (1+x)/(1-5*x) ).
%K A169403 nonn
%O A169403 0,2
%A A169403 _John Cannon_ and _N. J. A. Sloane_, Dec 03 2009