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A166368 Number of reduced words of length n in Coxeter group on 10 generators S_i with relations (S_i)^2 = (S_i S_j)^11 = I.

%I A166368 #22 Jul 23 2024 20:29:29
%S A166368 1,10,90,810,7290,65610,590490,5314410,47829690,430467210,3874204890,
%T A166368 34867843965,313810595280,2824295353920,25418658152880,
%U A166368 228767923084320,2058911305134480,18530201722590720,166771815290740080
%N A166368 Number of reduced words of length n in Coxeter group on 10 generators S_i with relations (S_i)^2 = (S_i S_j)^11 = I.
%C A166368 The initial terms coincide with those of A003952, although the two sequences are eventually different.
%C A166368 Computed with MAGMA using commands similar to those used to compute A154638.
%H A166368 G. C. Greubel, <a href="/A166368/b166368.txt">Table of n, a(n) for n = 0..500</a>
%H A166368 <a href="/index/Rec#order_11">Index entries for linear recurrences with constant coefficients</a>, signature (8,8,8,8,8,8,8,8,8,8,-36).
%F A166368 G.f.: (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)/(36*t^11 - 8*t^10 - 8*t^9 - 8*t^8 - 8*t^7 - 8*t^6 - 8*t^5 - 8*t^4 - 8*t^3 - 8*t^2 - 8*t + 1).
%F A166368 From _G. C. Greubel_, Jul 23 2024: (Start)
%F A166368 a(n) = 8*Sum_{j=1..10} a(n-j) - 36*a(n-11).
%F A166368 G.f.: (1+t)*(1 - t^11)/(1 - 9*t + 44*t^11 - 36*t^12). (End)
%p A166368 seq(coeff(series((1+t)*(1-t^11)/(1-9*t+44*t^11-36*t^12), t, n+1), t, n), n = 0 .. 30); # _G. C. Greubel_, Mar 14 2020
%t A166368 CoefficientList[Series[(1+t)*(1-t^11)/(1-9*t+44*t^11-36*t^12), {t,0,30}], t] (* _G. C. Greubel_, May 10 2016 *)
%t A166368 coxG[{11, 36, -8}] (* The coxG program is in A169452 *) (* _G. C. Greubel_, Mar 14 2020 *)
%o A166368 (Sage)
%o A166368 def A166368_list(prec):
%o A166368     P.<t> = PowerSeriesRing(ZZ, prec)
%o A166368     return P( (1+t)*(1-t^11)/(1-9*t+44*t^11-36*t^12) ).list()
%o A166368 A166368_list(30) # _G. C. Greubel_, Mar 14 2020
%o A166368 (Magma)
%o A166368 R<x>:=PowerSeriesRing(Integers(), 30);
%o A166368 Coefficients(R!( (1+x)*(1-x^11)/(1-9*x+44*x^16-36*x^17) )); // _G. C. Greubel_, Jul 23 2024
%Y A166368 Cf. A003952, A154638, A169452.
%K A166368 nonn
%O A166368 0,2
%A A166368 _John Cannon_ and _N. J. A. Sloane_, Dec 03 2009