A154638 -id:A154638 - cp's OEIS frontend

A167919 Number of reduced words of length n in Coxeter group on 13 generators S_i with relations (S_i)^2 = (S_i S_j)^16 = I.

1, 13, 156, 1872, 22464, 269568, 3234816, 38817792, 465813504, 5589762048, 67077144576, 804925734912, 9659108818944, 115909305827328, 1390911669927936, 16690940039135232, 200291280469622706

Offset: 0

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

nonn

The initial terms coincide with those of A170732, although the two sequences are eventually different.

Computed with MAGMA using commands similar to those used to compute A154638.

G. C. Greubel, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,-66).

Cf. A154638, A169452, A170732.

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

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1+x)*(1-x^16)/(1-12*x+77*x^16-66*x^17) )); // G. C. Greubel, Sep 13 2023

CoefficientList[Series[(1+t)*(1-t^16)/(1-12*t+77*t^16-66*t^17), {t, 0, 50}], t] (* G. C. Greubel, Jul 01 2016; Sep 13 2023 *)
coxG[{16,66,-11}] (* The coxG program is at A169452 *) (* Harvey P. Dale, Mar 28 2018 *)

def A167919_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1+x)*(1-x^16)/(1-12*x+77*x^16-66*x^17) ).list()
A167919_list(40) # G. C. Greubel, Sep 13 2023

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)/( 66*t^16 - 11*t^15 - 11*t^14 - 11*t^13 - 11*t^12 - 11*t^11 - 11*t^10 - 11*t^9 - 11*t^8 - 11*t^7 - 11*t^6 - 11*t^5 - 11*t^4 - 11*t^3 - 11*t^2 - 11*t + 1).
From G. C. Greubel, Sep 13 2023: (Start)
G.f.: (1+t)*(1-t^16)/(1 - 12*t + 77*t^16 - 66*t^17).
a(n) = 11*Sum_{j=1..15} a(n-j) - 66*a(n-16). (End)

A167922 Number of reduced words of length n in Coxeter group on 14 generators S_i with relations (S_i)^2 = (S_i S_j)^16 = I.

Original entry on oeis.org

1, 14, 182, 2366, 30758, 399854, 5198102, 67575326, 878479238, 11420230094, 148462991222, 1930018885886, 25090245516518, 326173191714734, 4240251492291542, 55123269399790046, 716602502197270507

Offset: 0

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

nonn

The initial terms coincide with those of A170733, although the two sequences are eventually different.

Computed with MAGMA using commands similar to those used to compute A154638.

G. C. Greubel, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,-78).

Cf. A154638, A169452, A170733.

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

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1+x)*(1-x^16)/(1-13*x+90*x^16-78*x^17) )); // G. C. Greubel, Sep 13 2023

CoefficientList[Series[(1+t)*(1-t^16)/(1-13*t+90*t^16-78*t^17), {t, 0, 50}], t] (* G. C. Greubel, Jul 01 2016; Sep 13 2023 *)
coxG[{16, 78, -12, 40}] (* The coxG program is at A169452 *) (* G. C. Greubel, Sep 13 2023 *)

def A167922_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1+x)*(1-x^16)/(1-13*x+90*x^16-78*x^17) ).list()
A167922_list(40) # G. C. Greubel, Sep 13 2023

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)/( 78*t^16 - 12*t^15 - 12*t^14 - 12*t^13 - 12*t^12 - 12*t^11 - 12*t^10 - 12*t^9 - 12*t^8 - 12*t^7 - 12*t^6 - 12*t^5 - 12*t^4 - 12*t^3 - 12*t^2 - 12*t + 1).
From G. C. Greubel, Sep 13 2023: (Start)
G.f.: (1+t)*(1-t^16)/(1 - 13*t + 90*t^16 - 78*t^17).
a(n) = 12*Sum_{j=1..15} a(n-j) - 78*a(n-16). (End)

A167926 Number of reduced words of length n in Coxeter group on 17 generators S_i with relations (S_i)^2 = (S_i S_j)^16 = I.

Original entry on oeis.org

1, 17, 272, 4352, 69632, 1114112, 17825792, 285212672, 4563402752, 73014444032, 1168231104512, 18691697672192, 299067162755072, 4785074604081152, 76561193665298432, 1224979098644774912, 19599665578316398456

Offset: 0

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

nonn

The initial terms coincide with those of A170736, although the two sequences are eventually different.

Computed with MAGMA using commands similar to those used to compute A154638.

G. C. Greubel, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,-120).

Cf. A154638, A169452, A170736.

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

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1+x)*(1-x^16)/(1-16*x+135*x^16-120*x^17) )); // G. C. Greubel, Sep 10 2023

coxG[{16,120,-15}] (* The coxG program is at A169452 *) (* Harvey P. Dale, Nov 27 2015 *)
CoefficientList[Series[(1+t)*(1-t^16)/(1-16*t+135*t^16-120*t^17), {t, 0, 50}], t] (* G. C. Greubel, Jul 01 2016; Sep 10 2023 *)

def A167926_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1+x)*(1-x^16)/(1-16*x+135*x^16-120*x^17) ).list()
A167926_list(40) # G. C. Greubel, Sep 10 2023

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)/( 120*t^16 - 15*t^15 - 15*t^14 - 15*t^13 - 15*t^12 - 15*t^11 - 15*t^10 - 15*t^9 - 15*t^8 - 15*t^7 - 15*t^6 - 15*t^5 - 15*t^4 - 15*t^3 - 15*t^2 - 15*t + 1).
From G. C. Greubel, Sep 10 2023: (Start)
G.f.: (1+t)*(1-t^16)/(1 - 16*t + 135*t^16 - 120*t^17).
a(n) = 15*Sum_{j=1..15} a(n-j) - 120*a(n-16). (End)

A167927 Number of reduced words of length n in Coxeter group on 18 generators S_i with relations (S_i)^2 = (S_i S_j)^16 = I.

Original entry on oeis.org

1, 18, 306, 5202, 88434, 1503378, 25557426, 434476242, 7386096114, 125563633938, 2134581776946, 36287890208082, 616894133537394, 10487200270135698, 178282404592306866, 3030800878069216722, 51523614927176684121

Offset: 0

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

nonn

The initial terms coincide with those of A170737, although the two sequences are eventually different.

Computed with MAGMA using commands similar to those used to compute A154638.

G. C. Greubel, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,-136).

Cf. A154638, A169452, A170737.

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

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1+x)*(1-x^16)/(1-17*x+152*x^16-136*x^17) )); // G. C. Greubel, Sep 10 2023

CoefficientList[Series[(1+t)*(1-t^16)/(1-17*t+152*t^16-136*t^17), {t, 0, 50}], t] (* G. C. Greubel, Jul 01 2016; Sep 10 2023 *)
coxG[{16,136,-16}] (* The coxG program is at A169452 *) (* Harvey P. Dale, May 15 2022 *)

def A167927_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1+x)*(1-x^16)/(1-17*x+152*x^16-136*x^17) ).list()
A167927_list(40) # G. C. Greubel, Sep 10 2023

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)/( 136*t^16 - 16*t^15 - 16*t^14 - 16*t^13 - 16*t^12 - 16*t^11 - 16*t^10 - 16*t^9 - 16*t^8 - 16*t^7 - 16*t^6 - 16*t^5 - 16*t^4 - 16*t^3 - 16*t^2 - 16*t + 1).
From G. C. Greubel, Sep 10 2023: (Start)
G.f.: (1+t)*(1-t^16)/(1 - 17*t + 152*t^16 - 136*t^17).
a(n) = 16*Sum_{j=1..15} a(n-j) - 136*a(n-16). (End)

A167937 Number of reduced words of length n in Coxeter group on 23 generators S_i with relations (S_i)^2 = (S_i S_j)^16 = I.

Original entry on oeis.org

1, 23, 506, 11132, 244904, 5387888, 118533536, 2607737792, 57370231424, 1262145091328, 27767192009216, 610878224202752, 13439320932460544, 295665060514131968, 6504631331310903296, 143101889288839872512

Offset: 0

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

nonn

The initial terms coincide with those of A170742, although the two sequences are eventually different.

Computed with MAGMA using commands similar to those used to compute A154638.

G. C. Greubel, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,-231).

Cf. A154638, A169452, A170742.

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

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1+x)*(1-x^16)/(1-22*x+252*x^16-231*x^17) )); // G. C. Greubel, Sep 10 2023

CoefficientList[Series[(1+t)*(1-t^16)/(1-22*t+252*t^16-231*t^17), {t, 0, 50}], t] (* G. C. Greubel, Jul 01 2016; Sep 10 2023 *)
coxG[{16,231,-21}] (* The coxG program is at A169452 *) (* Harvey P. Dale, Dec 27 2016 *)

def A167937_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1+x)*(1-x^16)/(1-22*x+252*x^16-231*x^17) ).list()
A167937_list(40) # G. C. Greubel, Sep 10 2023

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)/( 231*t^16 - 21*t^15 - 21*t^14 - 21*t^13 - 21*t^12 - 21*t^11 - 21*t^10 - 21*t^9 - 21*t^8 - 21*t^7 - 21*t^6 - 21*t^5 - 21*t^4 - 21*t^3 - 21*t^2 - 21*t + 1).
From G. C. Greubel, Sep 10 2023: (Start)
G.f.: (1+t)*(1-t^16)/(1 - 22*t + 252*t^16 - 231*t^17).
a(n) = 21*Sum_{j=1..15} a(n-j) - 231*a(n-16). (End)

A154637 a(n) is the ratio of the sum of squares of the bends of the circles that are added in the n-th generation of Apollonian packing, to the sum of squares of the bends of the initial three circles.

Original entry on oeis.org

1, 2, 66, 1314, 26082, 517698, 10275714, 203961186, 4048396578, 80356048002, 1594975770306, 31658447262114, 628384017931362, 12472705016840898, 247568948283023874, 4913960850609954786, 97536510167350024098, 1935988320795170617602, 38427156885401362279746, 762735172745641733742114

Offset: 0

Colin Mallows, Jan 13 2009

For more references and links, see A189226.

			Starting with three circles with bends -1,2,2, the ssq is 9. The first derived generation has two circles, each with bend 3. So a(1) = (9+9)/9 = 2.

Colin Barker, Table of n, a(n) for n = 0..750
Colin Mallows, Growing Apollonian packings, J. Integer Sequences v.12, article 09.2.1 (2009).
Index entries for linear recurrences with constant coefficients, signature (20,-3).

For starting with four circles, see A137246. For sums of bends, see A135849 and A154636. For three dimensions, see A154638 - A154645.

Cf. also A189226, A189227.

CoefficientList[Series[(29 z^2 - 18 z + 1)/(3 z^2 - 20 z + 1), {z, 0, 100}], z] (* and *) LinearRecurrence[{20, -3}, {1, 2, 66}, 100] (* Vladimir Joseph Stephan Orlovsky, Jul 03 2011 *)

Vec((1-18*x+29*x^2)/(1-20*x+3*x^2) + O(x^30)) \\ Colin Barker, Nov 16 2016

G.f.: (1-18*x+29*x^2) / (1-20*x+3*x^2).
From Colin Barker, Nov 16 2016: (Start)
a(n) = ((133-13*sqrt(97))*(10+sqrt(97))^n - (10-sqrt(97))^n*(133+13*sqrt(97))) / (3*sqrt(97)) for n>0.
a(n) = 20*a(n-1) - 3*a(n-2) for n>2.
(End)

More terms from N. J. A. Sloane, Nov 22 2009

A169546 Number of reduced words of length n in Coxeter group on 5 generators S_i with relations (S_i)^2 = (S_i S_j)^35 = I.

Original entry on oeis.org

1, 5, 20, 80, 320, 1280, 5120, 20480, 81920, 327680, 1310720, 5242880, 20971520, 83886080, 335544320, 1342177280, 5368709120, 21474836480, 85899345920, 343597383680, 1374389534720, 5497558138880, 21990232555520, 87960930222080

Offset: 0

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

nonn

The initial terms coincide with those of A003947, although the two sequences are eventually different.

Computed with MAGMA using commands similar to those used to compute A154638.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, -6).

R:=PowerSeriesRing(Integers(), 25); Coefficients(R!( (1+x)*(1-x^35)/(1-4*x+9*x^35-6*x^36) )); // G. C. Greubel, Apr 25 2019

coxG[{35,6,-3,30}] (* The coxG program is at A169452 *) (* Harvey P. Dale, Jan 16 2015 *)
CoefficientList[Series[(1+x)*(1-x^35)/(1-4*x+9*x^35-6*x^36), {x, 0, 25}], x] (* G. C. Greubel, Apr 25 2019 *)

my(x='x+O('x^25)); Vec((1+x)*(1-x^35)/(1-4*x+9*x^35-6*x^36)) \\ G. C. Greubel, Apr 25 2019

((1+x)*(1-x^35)/(1-4*x+9*x^35-6*x^36)).series(x, 25).coefficients(x, sparse=False) # G. C. Greubel, Apr 25 2019

G.f.: (t^35 + 2*t^34 + 2*t^33 + 2*t^32 + 2*t^31 + 2*t^30 + 2*t^29 + 2*t^28 + 2*t^27 + 2*t^26 + 2*t^25 + 2*t^24 + 2*t^23 + 2*t^22 + 2*t^21 + 2*t^20 + 2*t^19 + 2*t^18 + 2*t^17 + 2*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)/(6*t^35 - 3*t^34 - 3*t^33 - 3*t^32 - 3*t^31 - 3*t^30 - 3*t^29 - 3*t^28 - 3*t^27 - 3*t^26 - 3*t^25 - 3*t^24 - 3*t^23 - 3*t^22 - 3*t^21 - 3*t^20 - 3*t^19 - 3*t^18 - 3*t^17 - 3*t^16 - 3*t^15 - 3*t^14 - 3*t^13 - 3*t^12 - 3*t^11 - 3*t^10 - 3*t^9 - 3*t^8 - 3*t^7 - 3*t^6 - 3*t^5 - 3*t^4 - 3*t^3 - 3*t^2 - 3*t + 1).

G.f.: (1+x)*(1-x^35)/(1 - 4*x + 9*x^35 - 6*x^36). - G. C. Greubel, Apr 25 2019

A163829 Number of reduced words of length n in Coxeter group on 48 generators S_i with relations (S_i)^2 = (S_i S_j)^5 = I.

Original entry on oeis.org

1, 48, 2256, 106032, 4983504, 234223560, 11008454304, 517394861664, 24317441438880, 1142914245838944, 53716710971646072, 2524673262335033136, 118659072125876564688, 5576949543463542381360, 262115366765585626863312

Offset: 0

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

nonn

The initial terms coincide with those of A170767, although the two sequences are eventually different.

Computed with MAGMA using commands similar to those used to compute A154638.

G. C. Greubel, Table of n, a(n) for n = 0..595
Index entries for linear recurrences with constant coefficients, signature (46, 46, 46, 46, -1081).

R:=PowerSeriesRing(Integers(), 20); Coefficients(R!( (1+x)*(1-x^5)/(1-47*x+1127*x^5-1081*x^6) )); // G. C. Greubel, Apr 25 2019

CoefficientList[Series[(1+x)*(1-x^5)/(1-47*x+1127*x^5-1081*x^6), {x, 0, 20}], x] (* G. C. Greubel, Aug 05 2017, modified Apr 25 2019 *)

my(x='x+O('x^20)); Vec((1+x)*(1-x^5)/(1-47*x+1127*x^5-1081*x^6)) \\ G. C. Greubel, Aug 05 2017, modified Apr 25 2019

((1+x)*(1-x^5)/(1-47*x+1127*x^5-1081*x^6)).series(x, 20).coefficients(x, sparse=False) # G. C. Greubel, Apr 25 2019

G.f.: (t^5 + 2*t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/(1081*t^5 - 46*t^4 - 46*t^3 - 46*t^2 - 46*t + 1).

G.f.: (1+x)*(1-x^5)/(1 -47*x +1127*x^5 -1081*x^6). - G. C. Greubel, Apr 25 2019

A163993 Number of reduced words of length n in Coxeter group on 25 generators S_i with relations (S_i)^2 = (S_i S_j)^6 = I.

Original entry on oeis.org

1, 25, 600, 14400, 345600, 8294400, 199065300, 4777560000, 114661267500, 2751866280000, 66044691360000, 1585070208000000, 38041627760729700, 912997692709095600, 21911911659905871900, 525885088676233035600

Offset: 0

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

nonn

The initial terms coincide with those of A170744, although the two sequences are eventually different.

Computed with MAGMA using commands similar to those used to compute A154638.

G. C. Greubel, Table of n, a(n) for n = 0..720
Index entries for linear recurrences with constant coefficients, signature (23, 23, 23, 23, 23, -276).

R:=PowerSeriesRing(Integers(), 20); Coefficients(R!( (1+x)*(1-x^7)/(1-24*x+299*x^6-276*x^7) )); // G. C. Greubel, Apr 25 2019

CoefficientList[Series[(1+x)*(1-x^7)/(1-24*x+299*x^6-276*x^7), {x,0,20}], x] (* G. C. Greubel, Aug 24 2017, modified Apr 25 2019 *)
coxG[{6,276,-23}] (* The coxG program is at A169452 *) (* Harvey P. Dale, Aug 02 2018 *)

my(x='x+O('x^20)); Vec((1+x)*(1-x^7)/(1-24*x+299*x^6-276*x^7)) \\ G. C. Greubel, Aug 24 2017, modified Apr 25 2019

((1+x)*(1-x^7)/(1-24*x+299*x^6-276*x^7)).series(x, 20).coefficients(x, sparse=False) # G. C. Greubel, Apr 25 2019

G.f.: (t^6 + 2*t^5 + 2*t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/(276*t^6 - 23*t^5 - 23*t^4 - 23*t^3 - 23*t^2 - 23*t + 1).

G.f.: (1+x)*(1-x^7)/(1 -24*x +299*x^6 -276*x^7). - G. C. Greubel, Apr 25 2019

A164091 Number of reduced words of length n in Coxeter group on 41 generators S_i with relations (S_i)^2 = (S_i S_j)^6 = I.

Original entry on oeis.org

1, 41, 1640, 65600, 2624000, 104960000, 4198399180, 167935934400, 6717436064820, 268697390145600, 10747893507936000, 429915656401920000, 17196622899456671580, 687864781713487950000, 27514585897949409744420

Offset: 0

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

nonn

The initial terms coincide with those of A170760, although the two sequences are eventually different.

Computed with MAGMA using commands similar to those used to compute A154638.

G. C. Greubel, Table of n, a(n) for n = 0..620
Index entries for linear recurrences with constant coefficients, signature (39, 39, 39, 39, 39, -780).

R:=PowerSeriesRing(Integers(), 20); Coefficients(R!( (1+x)*(1-x^6)/(1-40*x+819*x^6-780*x^7) )); // G. C. Greubel, Apr 25 2019

coxG[{6,780,-39}] (* The coxG program is at A169452 *) (* Harvey P. Dale, Apr 25 2015 *)
CoefficientList[Series[(1+x)*(1-x^6)/(1-40*x+819*x^6-780*x^7), {x,0,20}], x] (* G. C. Greubel, Apr 25 2019 *)

my(x='x+O('x^20)); Vec((1+x)*(1-x^6)/(1-40*x+819*x^6-780*x^7)) \\ G. C. Greubel, Apr 25 2019

((1+x)*(1-x^6)/(1-40*x+819*x^6-780*x^7)).series(x, 20).coefficients(x, sparse=False) # G. C. Greubel, Apr 25 2019

G.f.: (t^6 + 2*t^5 + 2*t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/(780*t^6 - 39*t^5 - 39*t^4 - 39*t^3 - 39*t^2 - 39*t + 1).

G.f.: (1+x)*(1-x^6)/(1 -40*x +819*x^6 -780*x^7). - G. C. Greubel, Apr 25 2019

cp's OEIS Frontend

A167919 Number of reduced words of length n in Coxeter group on 13 generators S_i with relations (S_i)^2 = (S_i S_j)^16 = I.

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A167922 Number of reduced words of length n in Coxeter group on 14 generators S_i with relations (S_i)^2 = (S_i S_j)^16 = I.

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A167926 Number of reduced words of length n in Coxeter group on 17 generators S_i with relations (S_i)^2 = (S_i S_j)^16 = I.

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A167927 Number of reduced words of length n in Coxeter group on 18 generators S_i with relations (S_i)^2 = (S_i S_j)^16 = I.

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A167937 Number of reduced words of length n in Coxeter group on 23 generators S_i with relations (S_i)^2 = (S_i S_j)^16 = I.

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A154637 a(n) is the ratio of the sum of squares of the bends of the circles that are added in the n-th generation of Apollonian packing, to the sum of squares of the bends of the initial three circles.

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A169546 Number of reduced words of length n in Coxeter group on 5 generators S_i with relations (S_i)^2 = (S_i S_j)^35 = I.

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