A167960 Number of reduced words of length n in Coxeter group on 44 generators S_i with relations (S_i)^2 = (S_i S_j)^16 = I.

1, 44, 1892, 81356, 3498308, 150427244, 6468371492, 278139974156, 11960018888708, 514280812214444, 22114074925221092, 950905221784506956, 40888924536733799108, 1758223755079553361644, 75603621468420794550692

Offset: 0

Author

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

Keywords

• nonn

Comments

The initial terms coincide with those of A170763, 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 (42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,-903).

Crossrefs

Cf. A154638, A169452, A170763.

Programs

Magma

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1+x)*(1+x^16)/(1-43*x+903*x^16-861*x^17) )); // G. C. Greubel, Apr 27 2023

Mathematica

CoefficientList[Series[(1+x)*(1+x^16)/(1-43*x+903*x^16-861*x^17), {x, 0, 50}], x] (* G. C. Greubel, Jul 02 2016; Apr 27 2023 *)
coxG[{16,903,-42}] (* The coxG program is at A169452 *) (* Harvey P. Dale, Jul 19 2021 *)

SageMath

def A167960_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1+x)*(1+x^16)/(1-43*x+903*x^16-861*x^17) ).list()
A167960_list(40) # G. C. Greubel, Apr 27 2023

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)/( 903*t^16 - 42*t^15 - 42*t^14 - 42*t^13 - 42*t^12 - 42*t^11 - 42*t^10 - 42*t^9 - 42*t^8 - 42*t^7 - 42*t^6 - 42*t^5 - 42*t^4 - 42*t^3 - 42*t^2 - 42*t + 1).

From G. C. Greubel, Apr 27 2023: (Start)

G.f.: (1 + x)*(1 + x^16)/(1 - 43*x + 903*x^16 - 861*x^17).

a(n) = 42*Sum_{k=1..m-1} a(n-k) - 903*a(n-m). (End)