A167958 Number of reduced words of length n in Coxeter group on 42 generators S_i with relations (S_i)^2 = (S_i S_j)^16 = I.

1, 42, 1722, 70602, 2894682, 118681962, 4865960442, 199504378122, 8179679503002, 335366859623082, 13750041244546362, 563751691026400842, 23113819332082434522, 947666592615379815402, 38854330297230572431482

Offset

0,2

Comments

The initial terms coincide with those of A170761, 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 (40,40,40,40,40,40,40,40,40,40,40,40,40,40,40,-820).

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)/( 820*t^16 - 40*t^15 - 40*t^14 - 40*t^13 - 40*t^12 - 40*t^11 - 40*t^10 - 40*t^9 - 40*t^8 - 40*t^7 - 40*t^6 - 40*t^5 - 40*t^4 - 40*t^3 - 40*t^2 - 40*t + 1).

From G. C. Greubel, Jul 14 2023: (Start)

G.f.: (1 + t)*(1 - t^16)/(1 - 41*t + 860*t^16 - 820*t^17).

a(n) = -820*a(n-16) + 40*Sum_{j=1..15} a(n-j). (End)

Mathematica

CoefficientList[Series[(1+t)*(1-t^16)/(1-41*t+860*t^16 -820*t^17), {t, 0, 40}], t] (* G. C. Greubel, Jul 02 2016; Jul 14 2023 *)
coxG[{16, 820, -40, 30}] (* The coxG program is at A169452 *) (* G. C. Greubel, Jul 14 2023 *)

Prog

(Magma) R<x>:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1+x)*(1-x^16)/(1-41*x+860*x^16-820*x^17) )); // G. C. Greubel, Jul 14 2023
(SageMath)
def A167958_list(prec):
    P.<x> = PowerSeriesRing(ZZ, prec)
    return P( (1+x)*(1-x^16)/(1-41*x+860*x^16-820*x^17) ).list()
A167958_list(40) # G. C. Greubel, Jul 14 2023

Crossrefs

Cf. A154638, A169452, A170761.

Sequence in context: A167095 A167599 A167848 * A168719 A168767 A168815

Adjacent sequences: A167955 A167956 A167957 * A167959 A167960 A167961

Keyword

nonn

Author

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

Status

approved