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A164618 Number of reduced words of length n in Coxeter group on 14 generators S_i with relations (S_i)^2 = (S_i S_j)^7 = I.

%I A164618 #17 Sep 08 2022 08:45:47
%S A164618 1,14,182,2366,30758,399854,5198102,67575235,878476872,11420184048,
%T A164618 148462193880,1930005936768,25090043590248,326170130032656,
%U A164618 4240206014105334,55122604391319192,716592897791781192
%N A164618 Number of reduced words of length n in Coxeter group on 14 generators S_i with relations (S_i)^2 = (S_i S_j)^7 = I.
%C A164618 The initial terms coincide with those of A170733, although the two sequences are eventually different.
%C A164618 Computed with MAGMA using commands similar to those used to compute A154638.
%H A164618 G. C. Greubel, <a href="/A164618/b164618.txt">Table of n, a(n) for n = 0..895</a>
%H A164618 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (12,12,12,12,12,12,-78).
%F A164618 G.f.: (t^7 + 2*t^6 + 2*t^5 + 2*t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/(78*t^7 - 12*t^6 - 12*t^5 - 12*t^4 - 12*t^3 - 12*t^2 - 12*t + 1).
%F A164618 a(n) = -78*a(n-7) + 12*Sum_{k=1..6} a(n-k). - _Wesley Ivan Hurt_, May 11 2021
%p A164618 seq(coeff(series((1+t)*(1-t^7)/(1-13*t+90*t^7-78*t^8), t, n+1), t, n), n = 0..20); # _G. C. Greubel_, Sep 15 2019
%t A164618 CoefficientList[Series[(1+t)*(1-t^7)/(1-13*t+90*t^7-78*t^8), {t,0,20}],t] (* _G. C. Greubel_, Aug 10 2017 *)
%t A164618 coxG[{7, 78, -12}] (* The coxG program is at A169452 *) (* _G. C. Greubel_, Sep 15 2019 *)
%o A164618 (PARI) my(t='t+O('t^20)); Vec((1+t)*(1-t^7)/(1-13*t+90*t^7-78*t^8)) \\ _G. C. Greubel_, Aug 10 2017
%o A164618 (Magma) R<t>:=PowerSeriesRing(Integers(), 20); Coefficients(R!( (1+t)*(1-t^7)/(1-13*t+90*t^7-78*t^8) )); // _G. C. Greubel_, Sep 15 2019
%o A164618 (Sage)
%o A164618 def A164618_list(prec):
%o A164618     P.<t> = PowerSeriesRing(ZZ, prec)
%o A164618     return P((1+t)*(1-t^7)/(1-13*t+90*t^7-78*t^8)).list()
%o A164618 A164618_list(20) # _G. C. Greubel_, Sep 15 2019
%o A164618 (GAP) a:=[14,182,2366,30758,399854,5198102,67575235];; for n in [8..20] do a[n]:=12*(a[n-1] +a[n-2]+a[n-3]+a[n-4]+a[n-5]+a[n-6]) -78*a[n-6]; od; Concatenation([1], a); # _G. C. Greubel_, Sep 15 2019
%K A164618 nonn
%O A164618 0,2
%A A164618 _John Cannon_ and _N. J. A. Sloane_, Dec 03 2009