A166442 - OEIS

A166442

Number of reduced words of length n in Coxeter group on 48 generators S_i with relations (S_i)^2 = (S_i S_j)^11 = I.

1, 48, 2256, 106032, 4983504, 234224688, 11008560336, 517402335792, 24317909782224, 1142941759764528, 53718262708932816, 2524758347319841224, 118663642324032484512, 5577191189229524281440, 262127985893787524168352

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OFFSET

0,2

COMMENTS

The initial terms coincide with those of A170767, 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 (46,46,46,46,46,46,46,46,46,46,-1081).

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)/(1081*t^11 - 46*t^10 - 46*t^9 - 46*t^8 - 46*t^7 - 46*t^6 - 46*t^5 - 46*t^4 - 46*t^3 - 46*t^2 - 46*t + 1).

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

a(n) = 46*Sum_{j=1..10} a(n-j) - 1081*a(n-11).

G.f.: (1+x)*(1-x^11)/(1 - 47*x + 1127*x^11 - 1081*x^12). (End)

MATHEMATICA

With[{num=Total[2t^Range[10] ]+t^11+1, den=Total[-46 t^Range[10]]+ 1081t^11+ 1}, CoefficientList[Series[num/den, {t, 0, 30}], t]] (* Harvey P. Dale, Jul 21 2011 *)

CoefficientList[Series[(x^11 + 2 x^10 + 2 x^9 + 2 x^8 + 2 x^7 + 2 x^6 + 2 x^5 + 2 x^4 + 2 x^3 + 2 x^2 + 2 x + 1)/(1081 x^11 - 46 x^10 - 46 x^9 - 46 x^8 - 46 x^7 - 46 x^6 - 46 x^5 - 46 x^4 - 46 x^3 - 46 x^2 - 46 x + 1), {x, 0, 20}], x] (* Vincenzo Librandi, May 10 2015 *)

coxG[{11, 1081, -46, 30}] (* The coxG program is at A169452 *) (* G. C. Greubel, 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(1081, 46, 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 A166442_list(prec):

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

return P( f(1081, 46, x) ).list()

A166442_list(30) # G. C. Greubel, Jul 26 2024