A265055 -id:A265055 - cp's OEIS frontend

A162740 Number of reduced words of length n in Coxeter group on 4 generators S_i with relations (S_i)^2 = (S_i S_j)^3 = I.

1, 4, 12, 30, 72, 168, 390, 900, 2076, 4782, 11016, 25368, 58422, 134532, 309804, 713406, 1642824, 3783048, 8711526, 20060676, 46195260, 106377294, 244963080, 564094968, 1298984214, 2991269124, 6888221772, 15862029150, 36526694472, 84112781928, 193692865350

Offset: 0

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

The initial terms coincide with those of A003946, although the two sequences are eventually different.
Computed with MAGMA using commands similar to those used to compute A154638.
From Bruno Berselli, Dec 28 2015: (Start)
Also, expansion of b(2)*b(3)/(1 - 2*x - 2*x^2 + 3*x^3), where b(k) = (1-x^k)/(1-x).
This is also the Poincaré series [or Poincare series] for the quasi-Lannér diagram QL4_22 - see Table 7.8 in Maxim Chapovalov, Dimitry Leites and Rafael Stekolshchik (2009), or equivalently Table 6 in the shorter version, Maxim Chapovalov, Dimitry Leites and Rafael Stekolshchik (2010).
(End)

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Maxim Chapovalov, Dimitry Leites, and Rafael Stekolshchik, The Poincaré series [or Poincare series] of the hyperbolic Coxeter groups with finite volume of fundamental domains, arXiv:0906.1596 [math.RT], 2009, page 31.
Maxim Chapovalov, Dimitry Leites, and Rafael Stekolshchik, The Poincaré series [or Poincare series] of the hyperbolic Coxeter groups with finite volume of fundamental domains, Journal of Nonlinear Mathematical Physics 17.supp01 (2010), page 186.
Index entries for linear recurrences with constant coefficients, signature (2,2,-3).

Cf. similar sequences listed in A265055.

m:=40; R:=PowerSeriesRing(Integers(), m); b:=func; Coefficients(R!(b(2)*b(3)/(1-2*x-2*x^2+3*x^3))); // Bruno Berselli, Dec 28 2015 - see Chapovalov et al.

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1+x)*(1-x^3)/(1-3*x+5*x^3-3*x^4) )); // G. C. Greubel, Apr 25 2019

CoefficientList[Series[(x^3+2x^2+2x+1)/(3x^3-2x^2-2x+1), {x, 0, 40}], x ] (* Vincenzo Librandi, Apr 29 2014 *)
coxG[{3, 3, -2, 40}] (* The coxG program is at A169452 *) (* G. C. Greubel, Apr 25 2019 *)

my(x='x+O('x^40)); Vec((1+x)*(1-x^3)/(1-3*x+5*x^3-3*x^4)) \\ G. C. Greubel, Apr 25 2019

((1+x)*(1-x^3)/(1-3*x+5*x^3-3*x^4)).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, Apr 25 2019

G.f.: (x^3 + 2*x^2 + 2*x + 1)/(3*x^3 - 2*x^2 - 2*x + 1).
From Bruno Berselli, Dec 28 2015: (Start)
a(n) = 2*a(n-1) + 2*a(n-2) - 3*a(n-3) for n>3.
a(n) = -2 + ((-7+2*sqrt(13))*(1-sqrt(13))^n + (7+2*sqrt(13))*(1+sqrt(13))^n)/(3*sqrt(13)*2^(n-1)) for n>0. (End)
G.f.: (1+x)*(1-x^3)/(1 -3*x +5*x^3 -3*x^4). - G. C. Greubel, Apr 25 2019

A264079 Expansion of b(2)b(6)/(1 - 2x + x^3 - x^4 - x^5 + 2*x^6), where b(k) = (1-x^k)/(1-x).

Original entry on oeis.org

1, 4, 10, 21, 41, 79, 150, 282, 527, 982, 1829, 3405, 6337, 11790, 21932, 40797, 75888, 141161, 262573, 488407, 908474, 1689830, 3143211, 5846606, 10875117, 20228513, 37626513, 69988066, 130182920, 242149745, 450416216, 837807065, 1558382345, 2898705007, 5391803070

Offset: 0

Bruno Berselli, Dec 28 2015

This is the Poincaré series [or Poincare series] for the quasi-Lannér diagram QL4_20 - see Table 7.8 in Maxim Chapovalov, Dimitry Leites and Rafael Stekolshchik (2009), or equivalently Table 6 in the shorter version, Maxim Chapovalov, Dimitry Leites and Rafael Stekolshchik (2010).

Colin Barker, Table of n, a(n) for n = 0..1000
Maxim Chapovalov, Dimitry Leites, and Rafael Stekolshchik, The Poincaré series [or Poincare series] of the hyperbolic Coxeter groups with finite volume of fundamental domains, arXiv:0906.1596 [math.RT], 2009, page 31.
Maxim Chapovalov, Dimitry Leites, and Rafael Stekolshchik, The Poincaré series [or Poincare series] of the hyperbolic Coxeter groups with finite volume of fundamental domains, Journal of Nonlinear Mathematical Physics, Volume 17, Supplement 1 (2010), page 186.
Index entries for linear recurrences with constant coefficients, signature (2,0,-1,1,1,-2).

Cf. similar sequences listed in A265055.

/* By definition: */ m:=40; R:=PowerSeriesRing(Integers(), m); b:=func; Coefficients(R!(b(2)*b(6)/(1-2*x+x^3-x^4-x^5+2*x^6)));

CoefficientList[Series[(1 + x)^2 (1 - x + x^2) (1 + x + x^2)/((1 - x) (1 - x - x^2 - x^4 - 2 x^5)), {x, 0, 40}], x]
LinearRecurrence[{2,0,-1,1,1,-2},{1,4,10,21,41,79,150},40] (* Harvey P. Dale, Jan 18 2021 *)

G.f.: (1 + x)^2*(1 - x + x^2)*(1 + x + x^2)/((1 - x)*(1 - x - x^2 - x^4 - 2*x^5)).

a(n) = 2*a(n-1) - a(n-3) + a(n-4) + a(n-5) - 2*a(n-6) for n>6.

A266333 G.f. = b(2)b(4)b(6)/(x^8+x^6-x^5-x^3-x+1), where b(k) = (1-x^k)/(1-x).

Original entry on oeis.org

1, 4, 9, 17, 29, 47, 74, 113, 170, 253, 375, 555, 818, 1203, 1767, 2594, 3807, 5584, 8188, 12004, 17597, 25795, 37809, 55416, 81220, 119038, 174464, 255694, 374742, 549215, 804918, 1179670, 1728895, 2533823, 3713502, 5442406, 7976239, 11689751, 17132167

Offset: 0

Alois P. Heinz, Dec 27 2015

This is the Poincaré series [or Poincare series] for the quasi-Lannér diagram QL4_2 - see Tables 7.6, 7.7 and 7.8 in Maxim Chapovalov, Dimitry Leites and Rafael Stekolshchik (2009), or equivalently Tables 5 and 6 in the shorter version, Maxim Chapovalov, Dimitry Leites and Rafael Stekolshchik (2010).

Colin Barker, Table of n, a(n) for n = 0..1000
Maxim Chapovalov, Dimitry Leites, and Rafael Stekolshchik, The Poincaré series [or Poincare series] of the hyperbolic Coxeter groups with finite volume of fundamental domains, arXiv:0906.1596 [math.RT], 2009.
Maxim Chapovalov, Dimitry Leites, and Rafael Stekolshchik, The Poincaré series [or Poincare series] of the hyperbolic Coxeter groups with finite volume of fundamental domains, Journal of Nonlinear Mathematical Physics 17.supp01 (2010), 169-215.
Index entries for linear recurrences with constant coefficients, signature (1,0,1,0,1,-1,0,-1).

Cf. similar sequences listed in A265055.

/* By definition: */ m:=40; R:=PowerSeriesRing(Integers(), m); b:=func; Coefficients(R!(b(2)*b(4)*b(6)/(x^8+x^6-x^5-x^3-x+1))); // Bruno Berselli, Dec 29 2015

gf:= b(2)*b(4)*b(6)/(x^8+x^6-x^5-x^3-x+1):
b:= k->(1-x^k)/(1-x):
a:= n-> coeff(series(gf, x, n+1), x, n):
seq(a(n), n=0..40);

b[k_] := (1 - x^k)/(1 - x); CoefficientList[Series[b[2] b[4] b[6]/(x^8 + x^6 - x^5 - x^3 - x + 1), {x, 0, 40}], x] (* Bruno Berselli, Dec 29 2015 *)

Vec((1+x)^3*(1-x+x^2)*(1+x^2)*(1+x+x^2) / ((1-x)*(1-x-x^3)*(1+x+x^2+x^3+x^4)) + O(x^50)) \\ Colin Barker, Dec 29 2015

G.f.: (1+x)^3*(1-x+x^2)*(1+x^2)*(1+x+x^2) / ((1-x)*(1-x-x^3)*(1+x+x^2+x^3+x^4)). - Colin Barker, Dec 29 2015

A266334 G.f. = b(2)b(6)b(10)/(x^14+x^12-x^5-x^3-x+1), where b(k) = (1-x^k)/(1-x).

Original entry on oeis.org

1, 4, 9, 17, 30, 51, 84, 135, 215, 341, 538, 846, 1328, 2082, 3262, 5108, 7997, 12519, 19595, 30668, 47996, 75112, 117546, 183950, 287864, 450478, 704950, 1103170, 1726339, 2701526, 4227582, 6615684, 10352789, 16200930, 25352598, 39673907, 62085111, 97156070

Offset: 0

Alois P. Heinz, Dec 27 2015

This is the Poincaré series [or Poincare series] for the quasi-Lannér diagram QL4_3 - see Tables 7.6, 7.7 and 7.8 in Maxim Chapovalov, Dimitry Leites and Rafael Stekolshchik (2009), or equivalently Tables 5 and 6 in the shorter version, Maxim Chapovalov, Dimitry Leites and Rafael Stekolshchik (2010).

Colin Barker, Table of n, a(n) for n = 0..1000
Maxim Chapovalov, Dimitry Leites, and Rafael Stekolshchik, The Poincaré series [or Poincare series] of the hyperbolic Coxeter groups with finite volume of fundamental domains, arXiv:0906.1596 [math.RT], 2009.
Maxim Chapovalov, Dimitry Leites, and Rafael Stekolshchik, The Poincaré series [or Poincare series] of the hyperbolic Coxeter groups with finite volume of fundamental domains, Journal of Nonlinear Mathematical Physics 17.supp01 (2010), 169-215.
Index entries for linear recurrences with constant coefficients, signature (1,0,1,0,1,0,0,0,0,0,0,-1,0,-1).

Cf. similar sequences listed in A265055.

/* By definition: */ m:=40; R:=PowerSeriesRing(Integers(), m); b:=func; Coefficients(R!(b(2)*b(6)*b(10)/(x^14+x^12-x^5-x^3-x+1))); // Bruno Berselli, Dec 29 2015

gf:= b(2)*b(6)*b(10)/(x^14+x^12-x^5-x^3-x+1):
b:= k->(1-x^k)/(1-x):
a:= n-> coeff(series(gf, x, n+1), x, n):
seq(a(n), n=0..40);

b[k_] := (1 - x^k)/(1 - x); CoefficientList[Series[b[2] b[6] b[10]/(x^14 + x^12 - x^5 - x^3 - x + 1), {x, 0, 40}], x] (* Bruno Berselli, Dec 29 2015 *)

A266335 G.f. = b(2)^2*b(6)/(x^7+x^6-x^5-x^2-x+1), where b(k) = (1-x^k)/(1-x).

Original entry on oeis.org

1, 4, 9, 17, 30, 52, 88, 145, 237, 386, 628, 1020, 1653, 2677, 4334, 7016, 11356, 18377, 29737, 48118, 77860, 125984, 203849, 329837, 533690, 863532, 1397228, 2260765, 3657997, 5918766, 9576768, 15495540, 25072313, 40567857, 65640174, 106208036, 171848216

Offset: 0

Alois P. Heinz, Dec 27 2015

This is the Poincaré series [or Poincare series] for the quasi-Lannér diagram QL4_4 - see Tables 7.6, 7.7 and 7.8 in Maxim Chapovalov, Dimitry Leites and Rafael Stekolshchik (2009), or equivalently Tables 5 and 6 in the shorter version, Maxim Chapovalov, Dimitry Leites and Rafael Stekolshchik (2010).

Colin Barker, Table of n, a(n) for n = 0..1000
Maxim Chapovalov, Dimitry Leites, and Rafael Stekolshchik, The Poincaré series [or Poincare series] of the hyperbolic Coxeter groups with finite volume of fundamental domains, arXiv:0906.1596 [math.RT], 2009.
Maxim Chapovalov, Dimitry Leites, and Rafael Stekolshchik, The Poincaré series [or Poincare series] of the hyperbolic Coxeter groups with finite volume of fundamental domains, Journal of Nonlinear Mathematical Physics 17.supp01 (2010), 169-215.
Index entries for linear recurrences with constant coefficients, signature (1,1,0,0,1,-1,-1).

Cf. similar sequences listed in A265055.

/* By definition: */ m:=40; R:=PowerSeriesRing(Integers(), m); b:=func; Coefficients(R!(b(2)^2*b(6)/(x^7+x^6-x^5-x^2-x+1))); // Bruno Berselli, Dec 29 2015

gf:= b(2)^2*b(6)/(x^7+x^6-x^5-x^2-x+1):
b:= k->(1-x^k)/(1-x):
a:= n-> coeff(series(gf, x, n+1), x, n):
seq(a(n), n=0..40);

b[k_] := (1 - x^k)/(1 - x); CoefficientList[Series[b[2]^2 b[6]/(x^7 + x^6 - x^5 - x^2 - x + 1), {x, 0, 40}], x] (* Bruno Berselli, Dec 29 2015 *)
LinearRecurrence[{1,1,0,0,1,-1,-1},{1,4,9,17,30,52,88,145},40] (* Harvey P. Dale, Mar 23 2020 *)

A266336 G.f. = b(2)b(6)/(x^6-x^4+x^2-2x+1), where b(k) = (1-x^k)/(1-x).

Original entry on oeis.org

1, 4, 9, 16, 26, 42, 67, 104, 158, 238, 359, 542, 816, 1224, 1833, 2746, 4116, 6168, 9237, 13828, 20702, 30998, 46415, 69492, 104034, 155746, 233171, 349090, 522628, 782420, 1171349, 1753622, 2625352, 3930412, 5884193, 8809176, 13188162, 19743938, 29558555

Offset: 0

Alois P. Heinz, Dec 27 2015

This is the Poincaré series [or Poincare series] for the quasi-Lannér diagram QL4_5 - see Tables 7.6, 7.7 and 7.8 in Maxim Chapovalov, Dimitry Leites and Rafael Stekolshchik (2009), or equivalently Tables 5 and 6 in the shorter version, Maxim Chapovalov, Dimitry Leites and Rafael Stekolshchik (2010).

Colin Barker, Table of n, a(n) for n = 0..1000
Maxim Chapovalov, Dimitry Leites, and Rafael Stekolshchik, The Poincaré series [or Poincare series] of the hyperbolic Coxeter groups with finite volume of fundamental domains, arXiv:0906.1596 [math.RT], 2009.
Maxim Chapovalov, Dimitry Leites, and Rafael Stekolshchik, The Poincaré series [or Poincare series] of the hyperbolic Coxeter groups with finite volume of fundamental domains, Journal of Nonlinear Mathematical Physics 17.supp01 (2010), 169-215.
Index entries for linear recurrences with constant coefficients, signature (2,-1,0,1,0,-1).

Cf. similar sequences listed in A265055.

/* By definition: */ m:=40; R:=PowerSeriesRing(Integers(), m); b:=func; Coefficients(R!(b(2)*b(6)/(x^6-x^4+x^2-2*x+1))); // Bruno Berselli, Dec 29 2015

gf:= b(2)*b(6)/(x^6-x^4+x^2-2*x+1):
b:= k->(1-x^k)/(1-x):
a:= n-> coeff(series(gf, x, n+1), x, n):
seq(a(n), n=0..40);

b[k_] := (1 - x^k)/(1 - x); CoefficientList[Series[b[2] b[6]/(x^6 - x^4 + x^2 - 2 x + 1), {x, 0, 40}], x] (* Bruno Berselli, Dec 29 2015 *)

A266337 Expansion of b(3)b(4)/(1 - 2x + x^5), where b(k) = (1-x^k)/(1-x).

Original entry on oeis.org

1, 4, 11, 25, 52, 104, 204, 397, 769, 1486, 2868, 5532, 10667, 20565, 39644, 76420, 147308, 283949, 547333, 1055022, 2033624, 3919940, 7555931, 14564529, 28074036, 54114448, 104308956, 201061981, 387559433, 747044830, 1439975212, 2775641468

Offset: 0

Bruno Berselli, Dec 27 2015

This is the Poincaré series [or Poincare series] for the quasi-Lannér diagram QL4_21 - see Table 7.8 in Maxim Chapovalov, Dimitry Leites and Rafael Stekolshchik (2009), or equivalently Table 6 in the shorter version, Maxim Chapovalov, Dimitry Leites and Rafael Stekolshchik (2010).

Colin Barker, Table of n, a(n) for n = 0..1000
Maxim Chapovalov, Dimitry Leites, and Rafael Stekolshchik, The Poincaré series [or Poincare series] of the hyperbolic Coxeter groups with finite volume of fundamental domains, arXiv:0906.1596 [math.RT], 2009, page 31.
Maxim Chapovalov, Dimitry Leites, and Rafael Stekolshchik, The Poincaré series [or Poincare series] of the hyperbolic Coxeter groups with finite volume of fundamental domains, Journal of Nonlinear Mathematical Physics, Volume 17, Supplement 1 (2010), page 186.
Index entries for linear recurrences with constant coefficients, signature (2,0,0,0,-1).

Cf. similar sequences listed in A265055.

/* By definition: */ m:=40; R:=PowerSeriesRing(Integers(), m); b:=func; Coefficients(R!(b(3)*b(4)/(1-2*x+x^5)));

CoefficientList[Series[(1 + x) (1 + x^2) (1 + x + x^2)/((1 - x) (1 - x - x^2 - x^3 - x^4)), {x, 0, 40}], x]

G.f.: (1 + x)*(1 + x^2)*(1 + x + x^2)/((1 - x)*(1 - x - x^2 - x^3 - x^4)).

a(n) = 2*a(n-1) - a(n-5) for n>5.

A266338 G.f. = b(2)b(4)b(6)/(x^8-x^3-x+1), where b(k) = (1-x^k)/(1-x).

Original entry on oeis.org

1, 4, 9, 17, 29, 46, 70, 104, 152, 219, 314, 449, 639, 907, 1286, 1821, 2576, 3643, 5150, 7277, 10281, 14524, 20515, 28975, 40923, 57795, 81620, 115266, 162780, 229876, 324627, 458432, 647385, 914217, 1291029, 1823148, 2574585, 3635738, 5134259, 7250412

Offset: 0

Alois P. Heinz, Dec 27 2015

This is the Poincaré series [or Poincare series] for the quasi-Lannér diagram QL4_6 - see Tables 7.6, 7.7 and 7.8 in Maxim Chapovalov, Dimitry Leites and Rafael Stekolshchik (2009), or equivalently Tables 5 and 6 in the shorter version, Maxim Chapovalov, Dimitry Leites and Rafael Stekolshchik (2010).

Colin Barker, Table of n, a(n) for n = 0..1000
Maxim Chapovalov, Dimitry Leites, and Rafael Stekolshchik, The Poincaré series [or Poincare series] of the hyperbolic Coxeter groups with finite volume of fundamental domains, arXiv:0906.1596 [math.RT], 2009.
Maxim Chapovalov, Dimitry Leites, and Rafael Stekolshchik, The Poincaré series [or Poincare series] of the hyperbolic Coxeter groups with finite volume of fundamental domains, Journal of Nonlinear Mathematical Physics 17.supp01 (2010), 169-215.
Index entries for linear recurrences with constant coefficients, signature (1,0,1,0,0,0,0,-1).

Cf. similar sequences listed in A265055.

gf:= b(2)*b(4)*b(6)/(x^8-x^3-x+1):
b:= k->(1-x^k)/(1-x):
a:= n-> coeff(series(gf, x, n+1), x, n):
seq(a(n), n=0..40);

b[k_] := (1 - x^k)/(1 - x); CoefficientList[Series[b[2] b[4] b[6]/(x^8 - x^3 - x + 1), {x, 0, 40}], x] (* Bruno Berselli, Dec 28 2015 *)

A266339 G.f. = b(2)^2*b(4)/(x^5+x^4-x^3-x^2-x+1), where b(k) = (1-x^k)/(1-x).

Original entry on oeis.org

1, 4, 9, 18, 33, 56, 94, 156, 255, 416, 677, 1098, 1780, 2884, 4669, 7558, 12233, 19796, 32034, 51836, 83875, 135716, 219597, 355318, 574920, 930244, 1505169, 2435418, 3940593, 6376016, 10316614, 16692636, 27009255, 43701896, 70711157, 114413058, 185124220

Offset: 0

Alois P. Heinz, Dec 27 2015

This is the Poincaré series [or Poincare series] for the quasi-Lannér diagram QL4_7 - see Tables 7.6, 7.7 and 7.8 in Maxim Chapovalov, Dimitry Leites and Rafael Stekolshchik (2009), or equivalently Tables 5 and 6 in the shorter version, Maxim Chapovalov, Dimitry Leites and Rafael Stekolshchik (2010).

Colin Barker, Table of n, a(n) for n = 0..1000
Maxim Chapovalov, Dimitry Leites, and Rafael Stekolshchik, The Poincaré series [or Poincare series] of the hyperbolic Coxeter groups with finite volume of fundamental domains, arXiv:0906.1596 [math.RT], 2009.
Maxim Chapovalov, Dimitry Leites, and Rafael Stekolshchik, The Poincaré series [or Poincare series] of the hyperbolic Coxeter groups with finite volume of fundamental domains, Journal of Nonlinear Mathematical Physics 17.supp01 (2010), 169-215.
Index entries for linear recurrences with constant coefficients, signature (1,1,1,-1,-1).

Cf. similar sequences listed in A265055.

gf:= b(2)^2*b(4)/(x^5+x^4-x^3-x^2-x+1):
b:= k->(1-x^k)/(1-x):
a:= n-> coeff(series(gf, x, n+1), x, n):
seq(a(n), n=0..40);

b[k_] := (1 - x^k)/(1 - x); CoefficientList[Series[b[2]^2 b[4]/(x^5 + x^4 - x^3 - x^2 - x + 1), {x, 0, 40}], x] (* Bruno Berselli, Dec 28 2015 *)

A266340 G.f. = b(2)b(4)b(6)/(x^8+x^6-x^5+x^4-2*x^3-x+1), where b(k) = (1-x^k)/(1-x).

Original entry on oeis.org

1, 4, 9, 18, 33, 56, 93, 151, 241, 383, 606, 956, 1506, 2369, 3724, 5852, 9193, 14439, 22676, 35609, 55916, 87801, 137865, 216473, 339899, 533696, 837986, 1315766, 2065951, 3243852, 5093330, 7997283, 12556917, 19716214, 30957365, 48607628, 76321141, 119835439

Offset: 0

Alois P. Heinz, Dec 27 2015

This is the Poincaré series [or Poincare series] for the quasi-Lannér diagram QL4_8 - see Tables 7.6, 7.7 and 7.8 in Maxim Chapovalov, Dimitry Leites and Rafael Stekolshchik (2009), or equivalently Tables 5 and 6 in the shorter version, Maxim Chapovalov, Dimitry Leites and Rafael Stekolshchik (2010).

Colin Barker, Table of n, a(n) for n = 0..1000
Maxim Chapovalov, Dimitry Leites, and Rafael Stekolshchik, The Poincaré series [or Poincare series] of the hyperbolic Coxeter groups with finite volume of fundamental domains, arXiv:0906.1596 [math.RT], 2009.
Maxim Chapovalov, Dimitry Leites, and Rafael Stekolshchik, The Poincaré series [or Poincare series] of the hyperbolic Coxeter groups with finite volume of fundamental domains, Journal of Nonlinear Mathematical Physics 17.supp01 (2010), 169-215.
Index entries for linear recurrences with constant coefficients, signature (2,-1,1,-1,1,-1).

Cf. similar sequences listed in A265055.

gf:= b(2)*b(4)*b(6)/(x^8+x^6-x^5+x^4-2*x^3-x+1):
b:= k->(1-x^k)/(1-x):
a:= n-> coeff(series(gf, x, n+1), x, n):
seq(a(n), n=0..40);

b[k_] := (1 - x^k)/(1 - x); CoefficientList[Series[b[2] b[4] b[6]/(x^8 + x^6 - x^5 + x^4 - 2 x^3 - x + 1), {x, 0, 40}], x] (* Bruno Berselli, Dec 28 2015 *)

cp's OEIS Frontend

A162740 Number of reduced words of length n in Coxeter group on 4 generators S_i with relations (S_i)^2 = (S_i S_j)^3 = I.

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A264079 Expansion of b(2)*b(6)/(1 - 2*x + x^3 - x^4 - x^5 + 2*x^6), where b(k) = (1-x^k)/(1-x).

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A266333 G.f. = b(2)*b(4)*b(6)/(x^8+x^6-x^5-x^3-x+1), where b(k) = (1-x^k)/(1-x).

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A266334 G.f. = b(2)*b(6)*b(10)/(x^14+x^12-x^5-x^3-x+1), where b(k) = (1-x^k)/(1-x).

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A266335 G.f. = b(2)^2*b(6)/(x^7+x^6-x^5-x^2-x+1), where b(k) = (1-x^k)/(1-x).

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A266336 G.f. = b(2)*b(6)/(x^6-x^4+x^2-2*x+1), where b(k) = (1-x^k)/(1-x).

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A266337 Expansion of b(3)*b(4)/(1 - 2*x + x^5), where b(k) = (1-x^k)/(1-x).

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A264079 Expansion of b(2)b(6)/(1 - 2x + x^3 - x^4 - x^5 + 2*x^6), where b(k) = (1-x^k)/(1-x).

A266333 G.f. = b(2)b(4)b(6)/(x^8+x^6-x^5-x^3-x+1), where b(k) = (1-x^k)/(1-x).

A266334 G.f. = b(2)b(6)b(10)/(x^14+x^12-x^5-x^3-x+1), where b(k) = (1-x^k)/(1-x).

A266336 G.f. = b(2)b(6)/(x^6-x^4+x^2-2x+1), where b(k) = (1-x^k)/(1-x).

A266337 Expansion of b(3)b(4)/(1 - 2x + x^5), where b(k) = (1-x^k)/(1-x).

A266338 G.f. = b(2)b(4)b(6)/(x^8-x^3-x+1), where b(k) = (1-x^k)/(1-x).

A266340 G.f. = b(2)b(4)b(6)/(x^8+x^6-x^5+x^4-2*x^3-x+1), where b(k) = (1-x^k)/(1-x).