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A167958
Number of reduced words of length n in Coxeter group on 42 generators S_i with relations (S_i)^2 = (S_i S_j)^16 = I.
1
1, 42, 1722, 70602, 2894682, 118681962, 4865960442, 199504378122, 8179679503002, 335366859623082, 13750041244546362, 563751691026400842, 23113819332082434522, 947666592615379815402, 38854330297230572431482
OFFSET
0,2
COMMENTS
The initial terms coincide with those of A170761, 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 (40,40,40,40,40,40,40,40,40,40,40,40,40,40,40,-820).
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)/( 820*t^16 - 40*t^15 - 40*t^14 - 40*t^13 - 40*t^12 - 40*t^11 - 40*t^10 - 40*t^9 - 40*t^8 - 40*t^7 - 40*t^6 - 40*t^5 - 40*t^4 - 40*t^3 - 40*t^2 - 40*t + 1).
From G. C. Greubel, Jul 14 2023: (Start)
G.f.: (1 + t)*(1 - t^16)/(1 - 41*t + 860*t^16 - 820*t^17).
a(n) = -820*a(n-16) + 40*Sum_{j=1..15} a(n-j). (End)
MATHEMATICA
CoefficientList[Series[(1+t)*(1-t^16)/(1-41*t+860*t^16 -820*t^17), {t, 0, 40}], t] (* G. C. Greubel, Jul 02 2016; Jul 14 2023 *)
coxG[{16, 820, -40, 30}] (* The coxG program is at A169452 *) (* G. C. Greubel, Jul 14 2023 *)
PROG
(Magma) R<x>:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1+x)*(1-x^16)/(1-41*x+860*x^16-820*x^17) )); // G. C. Greubel, Jul 14 2023
(SageMath)
def A167958_list(prec):
P.<x> = PowerSeriesRing(ZZ, prec)
return P( (1+x)*(1-x^16)/(1-41*x+860*x^16-820*x^17) ).list()
A167958_list(40) # G. C. Greubel, Jul 14 2023
CROSSREFS
KEYWORD
nonn
AUTHOR
John Cannon and N. J. A. Sloane, Dec 03 2009
STATUS
approved