A163977 Number of reduced words of length n in Coxeter group on 21 generators S_i with relations (S_i)^2 = (S_i S_j)^6 = I.

1, 21, 420, 8400, 168000, 3360000, 67199790, 1343991600, 26879748210, 537593288400, 10751832252000, 215035974720000, 4300706088043890, 86013853634593500, 1720271710182898110, 34405326953812846500

Offset

0,2

Comments

The initial terms coincide with those of A170740, 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..765

Index entries for linear recurrences with constant coefficients, signature (19,19,19,19,19,-190).

Formula

G.f.: (t^6 + 2*t^5 + 2*t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/(190*t^6 - 19*t^5 - 19*t^4 - 19*t^3 - 19*t^2 - 19*t + 1).

a(n) = -190*a(n-6) + 19*Sum_{k=1..5} a(n-k). - Wesley Ivan Hurt, May 11 2021

Maple

seq(coeff(series((1+t)*(1-t^6)/(1-20*t+209*t^6-190*t^7), t, n+1), t, n), n = 0 .. 30); # G. C. Greubel, Aug 11 2019

Mathematica

CoefficientList[Series[(1+t)*(1-t^6)/(1-20*t+209*t^6-190*t^7), {t, 0, 30}], t] (* G. C. Greubel, Aug 24 2017 *)
coxG[{6, 190, -19}] (* The coxG program is at A169452 *) (* G. C. Greubel, Aug 11 2019 *)

Prog

(PARI) my(t='t+O('t^30)); Vec((1+t)*(1-t^6)/(1-20*t+209*t^6-190*t^7)) \\ G. C. Greubel, Aug 24 2017
(Magma) R<t>:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1+t)*(1-t^6)/(1-20*t+209*t^6-190*t^7) )); // G. C. Greubel, Aug 11 2019
(Sage)
def A163977_list(prec):
    P.<t> = PowerSeriesRing(ZZ, prec)
    return P((1+t)*(1-t^6)/(1-20*t+209*t^6-190*t^7)).list()
A163977_list(30) # G. C. Greubel, Aug 11 2019
(GAP) a:=[21, 420, 8400, 168000, 3360000, 67199790];; for n in [7..30] do a[n]:=19*(a[n-1] +a[n-2]+a[n-3]+a[n-4]+a[n-5]) -190*a[n-6]; od; Concatenation([1], a); # G. C. Greubel, Aug 11 2019

Crossrefs

Sequence in context: A097833 A163145 A163503 * A164634 A164954 A165358

Adjacent sequences: A163974 A163975 A163976 * A163978 A163979 A163980

Keyword

nonn

Author

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

Status

approved