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A167908
Number of reduced words of length n in Coxeter group on 10 generators S_i with relations (S_i)^2 = (S_i S_j)^16 = I.
7
1, 10, 90, 810, 7290, 65610, 590490, 5314410, 47829690, 430467210, 3874204890, 34867844010, 313810596090, 2824295364810, 25418658283290, 228767924549610, 2058911320946445, 18530201888517600, 166771816996654800
OFFSET
0,2
COMMENTS
The initial terms coincide with those of A003952, although the two sequences are eventually different.
Computed with MAGMA using commands similar to those used to compute A154638.
LINKS
Index entries for linear recurrences with constant coefficients, signature (8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,-36).
FORMULA
G.f.: (t^16 + 2*t^15 + 2*t^14 + 2*t^13 + 2*t^12 + 2*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)/( 36*t^16 - 8*t^15 - 8*t^14 - 8*t^13 - 8*t^12 - 8*t^11 - 8*t^10 - 8*t^9 - 8*t^8 - 8*t^7 - 8*t^6 - 8*t^5 - 8*t^4 - 8*t^3 - 8*t^2 - 8*t + 1).
From G. C. Greubel, Jul 23 2024: (Start)
a(n) = 8*Sum_{j=1..15} a(n-j) - 36*a(n-16).
G.f.: (1+t)*(1 - t^16)/(1 - 9*t + 44*t^16 - 36*t^17). (End)
MATHEMATICA
With[{a=36, b=8}, CoefficientList[Series[(1+t)*(1-t^16)/(1-(b+1)*t +(a + b)*t^16 -a*t^17), {t, 0, 40}], t]] (* G. C. Greubel, Jul 01 2016; Jul 23 2024 *)
coxG[{16, 36, -8}] (* The coxG program is at A169452 *) (* Harvey P. Dale, Jun 04 2017 *)
PROG
(Magma)
R<x>:=PowerSeriesRing(Integers(), 30);
Coefficients(R!( (1+x)*(1-x^16)/(1-9*x+44*x^16-36*x^17) )); // G. C. Greubel, Jul 23 2024
(SageMath)
def A167908_list(prec):
P.<x> = PowerSeriesRing(ZZ, prec)
return P( (1+x)*(1-x^16)/(1-9*x+44*x^16-36*x^17) ).list()
A167908_list(30) # G. C. Greubel, Jul 23 2024
CROSSREFS
KEYWORD
nonn
AUTHOR
John Cannon and N. J. A. Sloane, Dec 03 2009
STATUS
approved