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A164069 Number of reduced words of length n in Coxeter group on 36 generators S_i with relations (S_i)^2 = (S_i S_j)^6 = I.

%I A164069 #17 Sep 08 2022 08:45:47
%S A164069 1,36,1260,44100,1543500,54022500,1890786870,66177518400,
%T A164069 2316212372880,81067406061600,2837358267534000,99307506301920000,
%U A164069 3475761563405646270,121651614218556733500,4257805080127526578980,149023128191211379381500
%N A164069 Number of reduced words of length n in Coxeter group on 36 generators S_i with relations (S_i)^2 = (S_i S_j)^6 = I.
%C A164069 The initial terms coincide with those of A170755, although the two sequences are eventually different.
%C A164069 Computed with MAGMA using commands similar to those used to compute A154638.
%H A164069 G. C. Greubel, <a href="/A164069/b164069.txt">Table of n, a(n) for n = 0..645</a>
%H A164069 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (34,34,34,34,34,-595).
%F A164069 G.f.: (t^6 + 2*t^5 + 2*t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/(595*t^6 - 34*t^5 - 34*t^4 - 34*t^3 - 34*t^2 - 34*t + 1).
%F A164069 a(n) = -595*a(n-6) + 34*Sum_{k=1..5} a(n-k). - _Wesley Ivan Hurt_, May 11 2021
%p A164069 seq(coeff(series((1+t)*(1-t^6)/(1-35*t+629*t^6-595*t^7), t, n+1), t, n), n = 0 .. 30); # _G. C. Greubel_, Aug 13 2019
%t A164069 CoefficientList[Series[(1+t)*(1-t^6)/(1-35*t+629*t^6-595*t^7), {t,0,30}], t] (* _G. C. Greubel_, Sep 09 2017 *)
%t A164069 coxG[{6, 595, -34}] (* The coxG program is at A169452 *) (* _G. C. Greubel_, Aug 13 2019 *)
%o A164069 (PARI) t='t+O('t^50); Vec((1+t)*(1-t^6)/(1-35*t+629*t^6-595*t^7)) \\ _G. C. Greubel_, Sep 09 2017
%o A164069 (Magma) R<t>:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1+t)*(1-t^6)/(1-35*t+629*t^6-595*t^7) )); // _G. C. Greubel_, Aug 13 2019
%o A164069 (Sage)
%o A164069 def A164069_list(prec):
%o A164069     P.<t> = PowerSeriesRing(ZZ, prec)
%o A164069     return P((1+t)*(1-t^6)/(1-35*t+629*t^6-595*t^7)).list()
%o A164069 A164069_list(30) # _G. C. Greubel_, Aug 13 2019
%o A164069 (GAP) a:=[36, 1260, 44100, 1543500, 54022500, 1890786870];; for n in [7..30] do a[n]:=34*(a[n-1] +a[n-2]+a[n-3]+a[n-4]+a[n-5]) -595*a[n-6]; od; Concatenation([1], a); # _G. C. Greubel_, Aug 13 2019
%K A164069 nonn
%O A164069 0,2
%A A164069 _John Cannon_ and _N. J. A. Sloane_, Dec 03 2009