A166441 - OEIS

A166441

Number of reduced words of length n in Coxeter group on 47 generators S_i with relations (S_i)^2 = (S_i S_j)^11 = I.

1, 47, 2162, 99452, 4574792, 210440432, 9680259872, 445291954112, 20483429889152, 942237774900992, 43342937645445632, 1993775131690497991, 91713656057762857860, 4218828178657089175245, 194066096218225996890780

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OFFSET

0,2

COMMENTS

The initial terms coincide with those of A170766, although the two sequences are eventually different.

Computed with MAGMA using commands similar to those used to compute A154638.

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..500

Index entries for linear recurrences with constant coefficients, signature (45,45,45,45,45,45,45,45,45,45,-1035).

FORMULA

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)/(1035*t^11 - 45*t^10 - 45*t^9 - 45*t^8 - 45*t^7 - 45*t^6 - 45*t^5 - 45*t^4 - 45*t^3 - 45*t^2 - 45*t + 1).

From G. C. Greubel, Jul 26 2024: (Start)

a(n) = 45*Sum_{j=1..10} a(n-j) - 1035*a(n-11).

G.f.: (1+x)*(1-x^11)/(1 - 46*x + 1080*x^11 - 1035*x^12). (End)

MATHEMATICA

coxG[{11, 1035, -45}] (* The coxG program is at A169452 *) (* Harvey P. Dale, Jan 16 2015 *)

With[{p=1035, q=45}, CoefficientList[Series[(1+t)*(1-t^11)/(1 - (q+1)*t + (p+q)*t^11 - p*t^12), {t, 0, 40}], t]] (* G. C. Greubel, May 14 2016; Jul 26 2024 *)

PROG

(Magma)

R<x>:=PowerSeriesRing(Integers(), 30);

f:= func< p, q, x | (1+x)*(1-x^11)/(1-(q+1)*x+(p+q)*x^11-p*x^12) >;

Coefficients(R!( f(1035, 45, x) )); // G. C. Greubel, Jul 26 2024

(SageMath)

def f(p, q, x): return (1+x)*(1-x^11)/(1-(q+1)*x+(p+q)*x^11-p*x^12)

def A166441_list(prec):

P.<x> = PowerSeriesRing(ZZ, prec)

return P( f(1035, 45, x) ).list()

A166441_list(30) # G. C. Greubel, Jul 26 2024

CROSSREFS