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A167941
Number of reduced words of length n in Coxeter group on 26 generators S_i with relations (S_i)^2 = (S_i S_j)^16 = I.
1
1, 26, 650, 16250, 406250, 10156250, 253906250, 6347656250, 158691406250, 3967285156250, 99182128906250, 2479553222656250, 61988830566406250, 1549720764160156250, 38743019104003906250, 968575477600097656250
OFFSET
0,2
COMMENTS
The initial terms coincide with those of A170745, 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 (24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,-300).
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)/( 300*t^16 - 24*t^15 - 24*t^14 - 24*t^13 - 24*t^12 - 24*t^11 - 24*t^10 - 24*t^9 - 24*t^8 - 24*t^7 - 24*t^6 - 24*t^5 - 24*t^4 - 24*t^3 - 24*t^2 - 24*t + 1).
From G. C. Greubel, Sep 08 2023: (Start)
G.f.: (1+t)*(1-t^16)/(1 - 25*t + 324*t^16 - 300*t^17).
a(n) = 24*Sum_{j=1..15} a(n-j) - 300*a(n-16). (End)
MATHEMATICA
coxG[{16, 300, -24}] (* The coxG program is at A169452 *) (* Harvey P. Dale, Jul 01 2021 *)
CoefficientList[Series[(1+t)*(1-t^16)/(1-25*t+324*t^16-300*t^17), {t, 0, 50}], t] (* G. C. Greubel, Sep 08 2023 *)
PROG
(Magma) R<x>:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1+x)*(1-x^16)/(1-25*x+324*x^16-300*x^17) )); // G. C. Greubel, Sep 08 2023
(SageMath)
def A167941_list(prec):
P.<x> = PowerSeriesRing(ZZ, prec)
return P( (1+x)*(1-x^16)/(1-25*x+324*x^16-300*x^17) ).list()
A167941_list(40) # G. C. Greubel, Sep 08 2023
KEYWORD
nonn
AUTHOR
John Cannon and N. J. A. Sloane, Dec 03 2009
STATUS
approved