A166432 - OEIS

A166432

Number of reduced words of length n in Coxeter group on 38 generators S_i with relations (S_i)^2 = (S_i S_j)^11 = I.

1, 38, 1406, 52022, 1924814, 71218118, 2635070366, 97497603542, 3607411331054, 133474219248998, 4938546112212926, 182726206151877559, 6760869627619443672, 250152176221918454160, 9255630520210947220872

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OFFSET

0,2

COMMENTS

The initial terms coincide with those of A170757, 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 (36,36,36,36,36,36,36,36,36,36,-666).

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)/(666*t^11 - 36*t^10 - 36*t^9 - 36*t^8 - 36*t^7 - 36*t^6 - 36*t^5 - 36*t^4 - 36*t^3 - 36*t^2 - 36*t + 1).

From G. C. Greubel, Jul 25 2024: (Start)

a(n) = 26*Sum_{j=1..10} a(n-j) - 351*a(n-11).

G.f.: (1+x)*(1-x^11)/(1 - 37*x + 702*x^11 - 666*x^12). (End)

MATHEMATICA

With[{p=666, q=36}, 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 25 2024 *)

coxG[{11, 666, -36}] (* The coxG program is at A169452 *) (* Harvey P. Dale, Sep 26 2016 *)

PROG

(Magma)

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

Coefficients(R!( (1+x)*(1-x^11)/(1-37*x+702*x^11-666*x^12) )); // G. C. Greubel, Jul 25 2024

(SageMath)

def A166432_list(prec):

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

return P( (1+x)*(1-x^11)/(1-37*x+702*x^11-666*x^12) ).list()

A166432_list(30) # G. C. Greubel, Jul 25 2024

CROSSREFS

Cf. A154638, A169452, A170757.