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A164667 Number of reduced words of length n in Coxeter group on 31 generators S_i with relations (S_i)^2 = (S_i S_j)^7 = I.

%I A164667 #10 Sep 08 2022 08:45:47
%S A164667 1,31,930,27900,837000,25110000,753300000,22598999535,677969972100,
%T A164667 20339098744965,610172949807900,18305188118005500,549155632253220000,
%U A164667 16474668628988250000,494240048711397215760,14827201156594414216125
%N A164667 Number of reduced words of length n in Coxeter group on 31 generators S_i with relations (S_i)^2 = (S_i S_j)^7 = I.
%C A164667 The initial terms coincide with those of A170750, although the two sequences are eventually different.
%C A164667 Computed with MAGMA using commands similar to those used to compute A154638.
%H A164667 G. C. Greubel, <a href="/A164667/b164667.txt">Table of n, a(n) for n = 0..670</a>
%H A164667 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (29,29,29,29,29,29,-435).
%F A164667 G.f.: (t^7 + 2*t^6 + 2*t^5 + 2*t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/(435*t^7 - 29*t^6 - 29*t^5 - 29*t^4 - 29*t^3 - 29*t^2 - 29*t + 1).
%p A164667 seq(coeff(series((1+t)*(1-t^7)/(1-30*t+464*t^7-435*t^8), t, n+1), t, n), n = 0 .. 20); # _G. C. Greubel_, Sep 15 2019
%t A164667 CoefficientList[Series[(1+t)*(1-t^7)/(1-30*t+464*t^7-435*t^8), {t, 0, 20}], t] (* _G. C. Greubel_, Sep 15 2019 *)
%t A164667 coxG[{7, 435, -29}] (* The coxG program is at A169452 *) (* _G. C. Greubel_, Sep 15 2019 *)
%o A164667 (PARI) my(t='t+O('t^20)); Vec((1+t)*(1-t^7)/(1-30*t+464*t^7-435*t^8)) \\ _G. C. Greubel_, Sep 15 2019
%o A164667 (Magma) R<t>:=PowerSeriesRing(Integers(), 20); Coefficients(R!( (1+t)*(1-t^7)/(1-30*t+464*t^7-435*t^8) )); // _G. C. Greubel_, Sep 15 2019
%o A164667 (Sage)
%o A164667 def A164667_list(prec):
%o A164667     P.<t> = PowerSeriesRing(ZZ, prec)
%o A164667     return P((1+t)*(1-t^7)/(1-30*t+464*t^7-435*t^8)).list()
%o A164667 A164667_list(20) # _G. C. Greubel_, Sep 15 2019
%o A164667 (GAP) a:=[31, 930, 27900, 837000, 25110000, 753300000, 22598999535];; for n in [8..20] do a[n]:=29*(a[n-1] +a[n-2]+a[n-3]+a[n-4]+a[n-5]+a[n-6]) -435*a[n-7]; od; Concatenation([1], a); # _G. C. Greubel_, Sep 15 2019
%K A164667 nonn
%O A164667 0,2
%A A164667 _John Cannon_ and _N. J. A. Sloane_, Dec 03 2009