A166463 - OEIS

A166463

Number of reduced words of length n in Coxeter group on 50 generators S_i with relations (S_i)^2 = (S_i S_j)^11 = I.

1, 50, 2450, 120050, 5882450, 288240050, 14123762450, 692064360050, 33911153642450, 1661646528480050, 81420679895522450, 3989613314880598825, 195491052429149282400, 9579061569028311897600, 469374016882387138922400

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OFFSET

0,2

COMMENTS

The initial terms coincide with those of A170769, 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 (48,48,48,48,48,48,48,48,48,48,-1176).

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)/(1176*t^11 - 48*t^10 - 48*t^9 - 48*t^8 - 48*t^7 - 48*t^6 - 48*t^5 - 48*t^4 - 48*t^3 - 48*t^2 - 48*t + 1).

From G. C. Greubel, Jul 27 2024: (Start)

a(n) = 48*Sum_{j=1..10} a(n-j) - 1176*a(n-11).

G.f.: (1+x)*(1-x^11)/(1 - 49*x + 1224*x^11 - 1176*x^12). (End)

MATHEMATICA

With[{p=1176, q=48}, 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 15 2016; Jul 27 2024 *)

coxG[{11, 1176, -48}] (* The coxG program is at A169452 *) (* Harvey P. Dale, Apr 29 2018 *)

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(1176, 48, x) )); // G. C. Greubel, Jul 27 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 A166463_list(prec):

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

return P( f(1176, 48, x) ).list()

A166463_list(30) # G. C. Greubel, Jul 27 2024