A165807 Number of reduced words of length n in Coxeter group on 12 generators S_i with relations (S_i)^2 = (S_i S_j)^10 = I.

1, 12, 132, 1452, 15972, 175692, 1932612, 21258732, 233846052, 2572306572, 28295372226, 311249093760, 3423740023440, 37661140170720, 414272540919600, 4556997939574080, 50126977219358160, 551396748137415840

Offset

0,2

Comments

The initial terms coincide with those of A003954, although the two sequences are eventually different.

First disagreement at index 10: a(10) = 28295372226, A003954(10) = 28295372292. - Klaus Brockhaus, Jun 14 2011

Computed with MAGMA using commands similar to those used to compute A154638.

Links

G. C. Greubel, Table of n, a(n) for n = 0..500

Index entries for linear recurrences with constant coefficients, signature (10,10,10,10,10,10,10,10,10,-55).

Formula

G.f.: (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^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).

Maple

seq(coeff(series((1+t)*(1-t^10)/(1-11*t+65*t^10-55*t^11), t, n+1), t, n), n = 0..20); # G. C. Greubel, Sep 23 2019

Mathematica

With[{num=Total[2t^Range[9]]+1+t^10, den=Total[-10 t^Range[9]]+1+ 55t^10}, CoefficientList[Series[num/den, {t, 0, 30}], t]] (* Harvey P. Dale, Jun 14 2011 *)
CoefficientList[Series[(1+t)*(1-t^10)/(1-11*t+65*t^10-55*t^11), {t, 0, 30}], t] (* or *) coxG[{10, 55, -10}] (* The coxG program is at A169452 *) (* G. C. Greubel, Sep 23 2019 *)

Prog

(PARI) my(t='t+O('t^20)); Vec((1+t)*(1-t^10)/(1-11*t+65*t^10-55*t^11)) \\ G. C. Greubel, Sep 23 2019
(Magma) R<t>:=PowerSeriesRing(Integers(), 20); Coefficients(R!( (1+t)*(1-t^10)/(1-11*t+65*t^10-55*t^11) )); // G. C. Greubel, Sep 23 2019
(Sage)
def A165807_list(prec):
    P.<t> = PowerSeriesRing(ZZ, prec)
    return P((1+t)*(1-t^10)/(1-11*t+65*t^10-55*t^11)).list()
A165807_list(20) # G. C. Greubel, Sep 23 2019
(GAP) a:=[12, 132, 1452, 15972, 175692, 1932612, 21258732, 233846052, 2572306572, 28295372226];; for n in [11..20] do a[n]:=10*Sum([1..9], j-> a[n-j]) -55*a[n-10]; od; Concatenation([1], a); # G. C. Greubel, Sep 23 2019

Crossrefs

Cf. A003954 (G.f.: (1+x)/(1-11*x)).

Sequence in context: A164601 A164781 A165266 * A166372 A166557 A166951

Adjacent sequences: A165804 A165805 A165806 * A165808 A165809 A165810

Keyword

nonn

Author

John Cannon and N. J. A. Sloane, Dec 03 2009

Status

approved