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A163958 Number of reduced words of length n in Coxeter group on 13 generators S_i with relations (S_i)^2 = (S_i S_j)^6 = I.

%I A163958 #17 Sep 08 2022 08:45:47
%S A163958 1,13,156,1872,22464,269568,3234738,38815920,465779886,5589224784,
%T A163958 67069091232,804809820672,9657486564726,115887063443580,
%U A163958 1390611458122458,16686937868633604,200238458992414128
%N A163958 Number of reduced words of length n in Coxeter group on 13 generators S_i with relations (S_i)^2 = (S_i S_j)^6 = I.
%C A163958 The initial terms coincide with those of A170732, although the two sequences are eventually different.
%C A163958 Computed with MAGMA using commands similar to those used to compute A154638.
%H A163958 G. C. Greubel, <a href="/A163958/b163958.txt">Table of n, a(n) for n = 0..920</a>
%H A163958 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (11,11,11,11,11,-66).
%F A163958 G.f.: (t^6 + 2*t^5 + 2*t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/(66*t^6 - 11*t^5 - 11*t^4 - 11*t^3 - 11*t^2 - 11*t + 1).
%F A163958 a(n) = -66*a(n-6) + 11*Sum_{k=1..5} a(n-k). - _Wesley Ivan Hurt_, May 11 2021
%p A163958 seq(coeff(series((1+t)*(1-t^6)/(1-12*t+77*t^6-66*t^7), t, n+1), t, n), n = 0 .. 30); # _G. C. Greubel_, Aug 11 2019
%t A163958 CoefficientList[Series[(1+t)*(1-t^6)/(1-12*t+77*t^6-66*t^7), {t,0,30}], t] (* _G. C. Greubel_, Aug 13 2017 *)
%t A163958 coxG[{6, 66, -11}] (* The coxG program is at A169452 *) (* _G. C. Greubel_, Aug 11 2019 *)
%o A163958 (PARI) my(t='t+O('t^30)); Vec((1+t)*(1-t^6)/(1-12*t+77*t^6-66*t^7)) \\ _G. C. Greubel_, Aug 13 2017
%o A163958 (Magma) R<t>:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1+t)*(1-t^6)/(1-12*t+77*t^6-66*t^7) )); // _G. C. Greubel_, Aug 11 2019
%o A163958 (Sage)
%o A163958 def A163878_list(prec):
%o A163958     P.<t> = PowerSeriesRing(ZZ, prec)
%o A163958     return P((1+t)*(1-t^6)/(1-12*t+77*t^6-66*t^7)).list()
%o A163958 A163878_list(30) # _G. C. Greubel_, Aug 11 2019
%o A163958 (GAP) a:=[13, 156, 1872, 22464, 269568, 3234738];; for n in [7..30] do a[n]:=11*(a[n-1] +a[n-2]+a[n-3]+a[n-4]+a[n-5]) -66*a[n-6]; od; Concatenation([1], a); # _G. C. Greubel_, Aug 11 2019
%K A163958 nonn
%O A163958 0,2
%A A163958 _John Cannon_ and _N. J. A. Sloane_, Dec 03 2009