A166437 Number of reduced words of length n in Coxeter group on 43 generators S_i with relations (S_i)^2 = (S_i S_j)^11 = I.

1, 43, 1806, 75852, 3185784, 133802928, 5619722976, 236028364992, 9913191329664, 416354035845888, 17486869505527296, 734448519232145529, 30846837807750074292, 1295567187925501528275, 54413821892870997324012

Offset

0,2

Comments

The initial terms coincide with those of A170762, 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 (41,41,41,41,41,41,41,41,41,41,-861).

Formula

G.f.: (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)/(861*t^11 - 41*t^10 - 41*t^9 - 41*t^8 - 41*t^7 - 41*t^6 - 41*t^5 - 41*t^4 - 41*t^3 - 41*t^2 - 41*t + 1).

a(n) = -861*a(n-11) + 41*Sum_{k=1..10} a(n-k). - Wesley Ivan Hurt, Mar 17 2023

G.f.: (1+x)*(1-x^11)/(1 - 42*x + 902*x^11 - 861*x^12). - G. C. Greubel, Jul 26 2024

Mathematica

With[{p=861, q=41}, CoefficientList[Series[(1+t)*(1-t^11)/(1 - (q+1)*t + (p+q)*t^11 - p*t^12), {t, 0, 40}], t]] (* G. C. Greubel, May 14 2016; Jul 26 2024 *)
coxG[{11, 861, -41, 30}] (* The coxG program is at A169452 *) (* G. C. Greubel, Jul 26 2024 *)

Prog

(Magma)
R<x>:=PowerSeriesRing(Integers(), 30);
f:= func< p, q, x | (1+x)*(1-x^11)/(1-(q+1)*x+(p+q)*x^11-p*x^12) >;
Coefficients(R!( f(861, 41, x) )); // G. C. Greubel, Jul 26 2024
(SageMath)
def f(p, q, x): return (1+x)*(1-x^11)/(1-(q+1)*x+(p+q)*x^11-p*x^12)
def A166437_list(prec):
    P.<x> = PowerSeriesRing(ZZ, prec)
    return P( f(861, 41, x) ).list()
A166437_list(30) # G. C. Greubel, Jul 26 2024

Crossrefs

Cf. A154638, A169452, A170762.

Sequence in context: A165175 A165694 A166233 * A166717 A167096 A167640

Adjacent sequences: A166434 A166435 A166436 * A166438 A166439 A166440

Keyword

nonn

Author

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

Status

approved