A166518 Number of reduced words of length n in Coxeter group on 7 generators S_i with relations (S_i)^2 = (S_i S_j)^12 = I.

1, 7, 42, 252, 1512, 9072, 54432, 326592, 1959552, 11757312, 70543872, 423263232, 2539579371, 15237476100, 91424855865, 548549130780, 3291294758220, 19747768390560, 118486609390800, 710919650629440, 4265517869484480

Offset: 0

Author

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

Keywords

• nonn
• easy

Comments

The initial terms coincide with those of A003949, although the two sequences are eventually different.

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 (5,5,5,5,5,5,5,5,5,5,5,-15).

Crossrefs

Cf. A003949, A154638, A169452.

Programs

Magma

R:=PowerSeriesRing(Integers(), 30);
f:= func< p,q,x | (1+x)*(1-x^12)/(1-(q+1)*x+(p+q)*x^12-p*x^13) >;
Coefficients(R!( f(15,5,x) )); // G. C. Greubel, Aug 03 2024

Mathematica

With[{p=15, q=5}, CoefficientList[Series[(1+t)*(1-t^12)/(1 - (q+1)*t + (p+q)*t^12 - p*t^13), {t,0,40}], t]] (* G. C. Greubel, May 15 2016; Aug 03 2024 *)
coxG[{12, 15, -5, 30}] (* The coxG program is at A169452 *) (* G. C. Greubel, Aug 03 2024 *)

PARI

Vec((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)/(15*t^12 - 5*t^11 - 5*t^10 - 5*t^9 - 5*t^8 - 5*t^7 - 5*t^6 - 5*t^5 - 5*t^4 - 5*t^3 - 5*t^2 - 5*t + 1)+O(t^99))

SageMath

def f(p,q,x): return (1+x)*(1-x^12)/(1-(q+1)*x+(p+q)*x^12-p*x^13)
def A166518_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( f(15,5,x) ).list()
A166518_list(30) # G. C. Greubel, Aug 03 2024

Formula

G.f.: (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)/(15*t^12 - 5*t^11 - 5*t^10 - 5*t^9 - 5*t^8 - 5*t^7 - 5*t^6 - 5*t^5 - 5*t^4 - 5*t^3 - 5*t^2 - 5*t + 1).

From G. C. Greubel, Aug 03 2024: (Start)

a(n) = 5*Sum_{j=1..11} a(n-j) - 15*a(n-12).

G.f.: (1+x)*(1-x^12)/(1 - 6*x + 20*x^12 - 15*x^13). (End)