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A167916
Number of reduced words of length n in Coxeter group on 12 generators S_i with relations (S_i)^2 = (S_i S_j)^16 = I.
7
1, 12, 132, 1452, 15972, 175692, 1932612, 21258732, 233846052, 2572306572, 28295372292, 311249095212, 3423740047332, 37661140520652, 414272545727172, 4556998002998892, 50126978032987746, 551396758362864480
OFFSET
0,2
COMMENTS
The initial terms coincide with those of A003954, 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 (10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,-55).
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)/( 55*t^16 - 10*t^15 - 10*t^14 - 10*t^13 - 10*t^12 - 10*t^11 - 10*t^10 - 10*t^9 - 10*t^8 - 10*t^7 - 10*t^6 - 10*t^5 - 10*t^4 - 10*t^3 - 10*t^2 - 10*t + 1).
From G. C. Greubel, Nov 10 2023: (Start)
a(n) = 10*Sum_{j=1..15} a(n-j) - 55*a(n-16).
G.f.: (1+x)*(1-x^16)/(1 - 11*x + 65*x^16 - 55*x^17). (End)
MATHEMATICA
CoefficientList[Series[(1+t)*(1-t^16)/(1-11*t+65*t^16-55*t^17), {t, 0, 50}], t] (* G. C. Greubel, Jul 01 2016; Nov 10 2023 *)
coxG[{16, 55, -10}] (* The coxG program is at A169452 *) (* G. C. Greubel, Nov 10 2023 *)
PROG
(Magma) R<x>:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1+x)*(1-x^16)/(1-11*x+65*x^16-55*x^17) )); // G. C. Greubel, Nov 10 2023
(SageMath)
def A167916_list(prec):
P.<x> = PowerSeriesRing(ZZ, prec)
return P( (1+x)*(1-x^16)/(1-11*x+65*x^16-55*x^17) ).list()
A167916_list(30) # G. C. Greubel, Nov 10 2023
CROSSREFS
KEYWORD
nonn
AUTHOR
John Cannon and N. J. A. Sloane, Dec 03 2009
STATUS
approved