A167941 Number of reduced words of length n in Coxeter group on 26 generators S_i with relations (S_i)^2 = (S_i S_j)^16 = I.

1, 26, 650, 16250, 406250, 10156250, 253906250, 6347656250, 158691406250, 3967285156250, 99182128906250, 2479553222656250, 61988830566406250, 1549720764160156250, 38743019104003906250, 968575477600097656250

Offset: 0

Author

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

Keywords

• nonn

Comments

The initial terms coincide with those of A170745, 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 (24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,-300).

Crossrefs

Cf. A154638, A169452, A170758.

Cf. A167881, A167882, A167896 - A167900, A167908, A167914, A167916, A167919, A167922, A167923, A167924, A167926, A167927, A167929, A167931, A167933, A167935, A167937, A167938, A167940, A167942 - A167947, A167949 - A167962, A167978, A167980, A167988, A167989.

Programs

Magma

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1+x)*(1-x^16)/(1-25*x+324*x^16-300*x^17) )); // G. C. Greubel, Sep 08 2023

Mathematica

coxG[{16,300,-24}] (* The coxG program is at A169452 *) (* Harvey P. Dale, Jul 01 2021 *)
CoefficientList[Series[(1+t)*(1-t^16)/(1-25*t+324*t^16-300*t^17), {t, 0, 50}], t] (* G. C. Greubel, Sep 08 2023 *)

SageMath

def A167941_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1+x)*(1-x^16)/(1-25*x+324*x^16-300*x^17) ).list()
A167941_list(40) # G. C. Greubel, Sep 08 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)/( 300*t^16 - 24*t^15 - 24*t^14 - 24*t^13 - 24*t^12 - 24*t^11 - 24*t^10 - 24*t^9 - 24*t^8 - 24*t^7 - 24*t^6 - 24*t^5 - 24*t^4 - 24*t^3 - 24*t^2 - 24*t + 1).

From G. C. Greubel, Sep 08 2023: (Start)

G.f.: (1+t)*(1-t^16)/(1 - 25*t + 324*t^16 - 300*t^17).

a(n) = 24*Sum_{j=1..15} a(n-j) - 300*a(n-16). (End)