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A162851
Number of reduced words of length n in Coxeter group on 37 generators S_i with relations (S_i)^2 = (S_i S_j)^3 = I.
1
1, 37, 1332, 47286, 1678320, 59557050, 2113447770, 74997827100, 2661373678950, 94441530616650, 3351353019273000, 118926143828399250, 4220214225380039250, 149758560520153357500, 5314333645481777358750, 188584492248078150341250
OFFSET
0,2
COMMENTS
The initial terms coincide with those of A170756, although the two sequences are eventually different.
Computed with MAGMA using commands similar to those used to compute A154638.
FORMULA
G.f.: (t^3 + 2*t^2 + 2*t + 1)/(630*t^3 - 35*t^2 - 35*t + 1).
G.f.: (1+x)*(1-x^3)/(1 - 36*x + 665*x^3 - 630*x^4). - G. C. Greubel, Apr 26 2019
a(n) = 35*a(n-1)+35*a(n-2)-630*a(n-3). - Wesley Ivan Hurt, May 05 2021
MATHEMATICA
CoefficientList[Series[(t^3+2*t^2+2*t+1)/(630*t^3-35*t^2-35*t+1), {t, 0, 20}], t] (* or *) LinearRecurrence[{35, 35, -630}, {1, 37, 1332}, 20] (* G. C. Greubel, Oct 24 2018 *)
coxG[{3, 630, -35}] (* The coxG program is at A169452 *) (* G. C. Greubel, Apr 26 2019 *)
PROG
(PARI) my(t='t+O('t^20)); Vec((t^3+2*t^2+2*t+1)/(630*t^3-35*t^2-35*t+1)) \\ G. C. Greubel, Oct 24 2018
(Magma) R<t>:=PowerSeriesRing(Integers(), 20); Coefficients(R!((t^3 +2*t^2+2*t+1)/(630*t^3-35*t^2-35*t+1))); // G. C. Greubel, Oct 24 2018
(Sage) ((1+x)*(1-x^3)/(1-36*x+665*x^3-630*x^4)).series(x, 20).coefficients(x, sparse=False) # G. C. Greubel, Apr 26 2019
(GAP) a:=[37, 1332, 47286];; for n in [4..20] do a[n]:=35*a[n-1]+ 35*a[n-2]-630*a[n-3]; od; Concatenation([1], a); # G. C. Greubel, Apr 26 2019
CROSSREFS
Sequence in context: A246461 A006062 A219420 * A163220 A163645 A164070
KEYWORD
nonn
AUTHOR
John Cannon and N. J. A. Sloane, Dec 03 2009
STATUS
approved