cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A167956 Number of reduced words of length n in Coxeter group on 40 generators S_i with relations (S_i)^2 = (S_i S_j)^16 = I.

Original entry on oeis.org

1, 40, 1560, 60840, 2372760, 92537640, 3608967960, 140749750440, 5489240267160, 214080370419240, 8349134446350360, 325616243407664040, 12699033492898897560, 495262306223057004840, 19315229942699223188760
Offset: 0

Views

Author

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

Keywords

Comments

The initial terms coincide with those of A170759, although the two sequences are eventually different.
Computed with Magma using commands similar to those used to compute A154638.

Links

Crossrefs

Cf. A154638, A169452, A170759.

Programs

  • Magma
    R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1+x)*(1-x^16)/(1-39*x+779*x^16-741*x^17) )); // G. C. Greubel, Jul 14 2023
  • Mathematica
    CoefficientList[Series[(1+t)*(1-t^16)/(1-39*t+779*t^16-741*t^17), {t, 0, 40}], t] (* G. C. Greubel, Jul 02 2016; Jul 14 2023 *)
coxG[{16,741,-38}] (* The coxG program is at A169452 *) (* Harvey P. Dale, Jun 22 2019 *)
  • SageMath
    def A167956_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1+x)*(1-x^16)/(1-39*x+779*x^16-741*x^17) ).list()
A167956_list(40) # G. C. Greubel, Jul 14 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)/( 741*t^16 - 38*t^15 - 38*t^14 - 38*t^13 - 38*t^12 - 38*t^11 - 38*t^10 - 38*t^9 - 38*t^8 - 38*t^7 - 38*t^6 - 38*t^5 - 38*t^4 - 38*t^3 - 38*t^2 - 38*t + 1).
From G. C. Greubel, Jul 14 2023: (Start)
G.f.: (1 + t)*(1 - t^16)/(1 - 39*t + 779*t^16 - 741*t^17).
a(n) = -741*a(n-16) + 38*Sum_{j=1..15} a(n-j). (End)

A168717 Number of reduced words of length n in Coxeter group on 40 generators S_i with relations (S_i)^2 = (S_i S_j)^17 = I.

Original entry on oeis.org

1, 40, 1560, 60840, 2372760, 92537640, 3608967960, 140749750440, 5489240267160, 214080370419240, 8349134446350360, 325616243407664040, 12699033492898897560, 495262306223057004840, 19315229942699223188760
Offset: 0

Views

Author

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

Keywords

Comments

The initial terms coincide with those of A170759, although the two sequences are eventually different.
First disagreement at index 17: a(17) = 1145760124970975220334053660, A170759(17) = 1145760124970975220334054440. - Klaus Brockhaus, Mar 28 2011
Computed with MAGMA using commands similar to those used to compute A154638.

Links

Crossrefs

Cf. A170759 (G.f.: (1+x)/(1-39*x)).

Programs

  • Mathematica
    CoefficientList[Series[(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)/(741*t^17 - 38*t^16 - 38*t^15 - 38*t^14 - 38*t^13 - 38*t^12 - 38*t^11 - 38*t^10 - 38*t^9 - 38*t^8 - 38*t^7 - 38*t^6 - 38*t^5 - 38*t^4 - 38*t^3 - 38*t^2 - 38*t + 1), {t,0,50}], t] (* G. C. Greubel, Aug 05 2016 *)

Formula

G.f.: (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)/(741*t^17 - 38*t^16 - 38*t^15 - 38*t^14 - 38*t^13 - 38*t^12 - 38*t^11 - 38*t^10 - 38*t^9 - 38*t^8 - 38*t^7 - 38*t^6 - 38*t^5 - 38*t^4 - 38*t^3 - 38*t^2 - 38*t + 1).

A168765 Number of reduced words of length n in Coxeter group on 40 generators S_i with relations (S_i)^2 = (S_i S_j)^18 = I.

Original entry on oeis.org

1, 40, 1560, 60840, 2372760, 92537640, 3608967960, 140749750440, 5489240267160, 214080370419240, 8349134446350360, 325616243407664040, 12699033492898897560, 495262306223057004840, 19315229942699223188760
Offset: 0

Views

Author

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

Keywords

Comments

The initial terms coincide with those of A170759, although the two sequences are eventually different.
First disagreement at index 18: a(18) = 44684644873868033593028122380, A170759(18) = 44684644873868033593028123160. - Klaus Brockhaus, Mar 26 2011
Computed with MAGMA using commands similar to those used to compute A154638.

Links

Crossrefs

Cf. A170759 (G.f.: (1+x)/(1-39*x)).

Programs

  • Mathematica
    With[{num=Total[2t^Range[17]]+t^18+1,den=Total[-38 t^Range[17]]+ 741t^18+ 1}, CoefficientList[Series[num/den,{t,0,20}],t]] (* Harvey P. Dale, Oct 31 2011 *)
CoefficientList[Series[(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)/(741*t^18 - 38*t^17 - 38*t^16 - 38*t^15 - 38*t^14 - 38*t^13 - 38*t^12 - 38*t^11 - 38*t^10 - 38*t^9 - 38*t^8 - 38*t^7 - 38*t^6 - 38*t^5 - 38*t^4 - 38*t^3 - 38*t^2 - 38*t + 1), {t, 0, 50}], t] (* G. C. Greubel, Aug 11 2016 *)

Formula

G.f.: (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)/(741*t^18 - 38*t^17 - 38*t^16 - 38*t^15 - 38*t^14 - 38*t^13 - 38*t^12 - 38*t^11 - 38*t^10 - 38*t^9 - 38*t^8 - 38*t^7 - 38*t^6 - 38*t^5 - 38*t^4 - 38*t^3 - 38*t^2 - 38*t + 1).

A168813 Number of reduced words of length n in Coxeter group on 40 generators S_i with relations (S_i)^2 = (S_i S_j)^19 = I.

Original entry on oeis.org

1, 40, 1560, 60840, 2372760, 92537640, 3608967960, 140749750440, 5489240267160, 214080370419240, 8349134446350360, 325616243407664040, 12699033492898897560, 495262306223057004840, 19315229942699223188760
Offset: 0

Views

Author

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

Keywords

Comments

The initial terms coincide with those of A170759, although the two sequences are eventually different.
First disagreement at index 19: a(19) = 1742701150080853310128096802460, A170759(19) = 1742701150080853310128096803240. - Klaus Brockhaus, Apr 01 2011
Computed with MAGMA using commands similar to those used to compute A154638.

Links

Crossrefs

Cf. A170759 (G.f.: (1+x)/(1-39*x)).

Programs

  • Mathematica
    CoefficientList[Series[(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)/(741*t^19 - 38*t^18 - 38*t^17 - 38*t^16 - 38*t^15 - 38*t^14 - 38*t^13 - 38*t^12 - 38*t^11 - 38*t^10 - 38*t^9 - 38*t^8 - 38*t^7 - 38*t^6 - 38*t^5 - 38*t^4 - 38*t^3 - 38*t^2 - 38*t + 1), {t, 0, 50}], t] (* G. C. Greubel, Aug 17 2016 *)
coxG[{19,741,-38}] (* The coxG program is at A169452 *) (* Harvey P. Dale, May 20 2021 *)

Formula

G.f.: (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)/(741*t^19 - 38*t^18 - 38*t^17 - 38*t^16 - 38*t^15 - 38*t^14 - 38*t^13 - 38*t^12 - 38*t^11 - 38*t^10 - 38*t^9 - 38*t^8 - 38*t^7 - 38*t^6 - 38*t^5 - 38*t^4 - 38*t^3 - 38*t^2 - 38*t + 1).

A170673 Number of reduced words of length n in Coxeter group on 40 generators S_i with relations (S_i)^2 = (S_i S_j)^49 = I.

Original entry on oeis.org

1, 40, 1560, 60840, 2372760, 92537640, 3608967960, 140749750440, 5489240267160, 214080370419240, 8349134446350360, 325616243407664040, 12699033492898897560, 495262306223057004840, 19315229942699223188760
Offset: 0

Views

Author

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

Keywords

Comments

The initial terms coincide with those of A170759, although the two sequences are eventually different.
Computed with MAGMA using commands similar to those used to compute A154638.

Links

Programs

  • Mathematica
    With[{num=Total[2t^Range[48]]+t^49+1,den=Total[-38 t^Range[48]]+741t^49+ 1},CoefficientList[Series[num/den,{t,0,20}],t]] (* Harvey P. Dale, Jul 21 2014 *)

Formula

G.f. (t^49 + 2*t^48 + 2*t^47 + 2*t^46 + 2*t^45 + 2*t^44 + 2*t^43 + 2*t^42 +
2*t^41 + 2*t^40 + 2*t^39 + 2*t^38 + 2*t^37 + 2*t^36 + 2*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)/(741*t^49 - 38*t^48 - 38*t^47 - 38*t^46 - 38*t^45 - 38*t^44 - 38*t^43
- 38*t^42 - 38*t^41 - 38*t^40 - 38*t^39 - 38*t^38 - 38*t^37 - 38*t^36 -
38*t^35 - 38*t^34 - 38*t^33 - 38*t^32 - 38*t^31 - 38*t^30 - 38*t^29 -
38*t^28 - 38*t^27 - 38*t^26 - 38*t^25 - 38*t^24 - 38*t^23 - 38*t^22 -
38*t^21 - 38*t^20 - 38*t^19 - 38*t^18 - 38*t^17 - 38*t^16 - 38*t^15 -
38*t^14 - 38*t^13 - 38*t^12 - 38*t^11 - 38*t^10 - 38*t^9 - 38*t^8 -
38*t^7 - 38*t^6 - 38*t^5 - 38*t^4 - 38*t^3 - 38*t^2 - 38*t + 1)

A170721 Number of reduced words of length n in Coxeter group on 40 generators S_i with relations (S_i)^2 = (S_i S_j)^50 = I.

Original entry on oeis.org

1, 40, 1560, 60840, 2372760, 92537640, 3608967960, 140749750440, 5489240267160, 214080370419240, 8349134446350360, 325616243407664040, 12699033492898897560, 495262306223057004840, 19315229942699223188760
Offset: 0

Views

Author

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

Keywords

Comments

The initial terms coincide with those of A170759, although the two sequences are eventually different.
Computed with MAGMA using commands similar to those used to compute A154638.
About the initial comment, first disagreement is at index 50 and the difference is 780. - Vincenzo Librandi, Dec 06 2012

Links

Programs

  • Mathematica
    With[{num = Total[2 t^Range[49]] + t^50 + 1, den = Total[-38  t^Range[49]] + 741 t^50 + 1}, CoefficientList[Series[num/den, {t, 0, 30}], t]] (* Vincenzo Librandi, Dec 06 2012 *)
coxG[{50,741,-38}] (* The coxG program is at A169452 *) (* Harvey P. Dale, Jan 03 2022 *)

Formula

G.f. (t^50 + 2*t^49 + 2*t^48 + 2*t^47 + 2*t^46 + 2*t^45 + 2*t^44 + 2*t^43 +
2*t^42 + 2*t^41 + 2*t^40 + 2*t^39 + 2*t^38 + 2*t^37 + 2*t^36 + 2*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)/(741*t^50 - 38*t^49 - 38*t^48 - 38*t^47 - 38*t^46 - 38*t^45 -
38*t^44 - 38*t^43 - 38*t^42 - 38*t^41 - 38*t^40 - 38*t^39 - 38*t^38 -
38*t^37 - 38*t^36 - 38*t^35 - 38*t^34 - 38*t^33 - 38*t^32 - 38*t^31 -
38*t^30 - 38*t^29 - 38*t^28 - 38*t^27 - 38*t^26 - 38*t^25 - 38*t^24 -
38*t^23 - 38*t^22 - 38*t^21 - 38*t^20 - 38*t^19 - 38*t^18 - 38*t^17 -
38*t^16 - 38*t^15 - 38*t^14 - 38*t^13 - 38*t^12 - 38*t^11 - 38*t^10 -
38*t^9 - 38*t^8 - 38*t^7 - 38*t^6 - 38*t^5 - 38*t^4 - 38*t^3 - 38*t^2 -
38*t + 1)

A164085 Number of reduced words of length n in Coxeter group on 40 generators S_i with relations (S_i)^2 = (S_i S_j)^6 = I.

Original entry on oeis.org

1, 40, 1560, 60840, 2372760, 92537640, 3608967180, 140749689600, 5489236708800, 214080185404800, 8349125427489600, 325615821343488000, 12699014289580990020, 495261450323895092760, 19315192390812032456040
Offset: 0

Views

Author

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

Keywords

Comments

The initial terms coincide with those of A170759, although the two sequences are eventually different.
Computed with MAGMA using commands similar to those used to compute A154638.

Links

Programs

Formula

G.f.: (t^6 + 2*t^5 + 2*t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/(741*t^6 - 38*t^5 - 38*t^4 - 38*t^3 - 38*t^2 - 38*t + 1).

A164684 Number of reduced words of length n in Coxeter group on 40 generators S_i with relations (S_i)^2 = (S_i S_j)^7 = I.

Original entry on oeis.org

1, 40, 1560, 60840, 2372760, 92537640, 3608967960, 140749749660, 5489240206320, 214080366860880, 8349134261335920, 325616234388803280, 12699033070834721520, 495262287019738489680, 19315229086799991370740
Offset: 0

Views

Author

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

Keywords

Comments

The initial terms coincide with those of A170759, although the two sequences are eventually different.
Computed with MAGMA using commands similar to those used to compute A154638.

Links

Programs

Formula

G.f.: (t^7 + 2*t^6 + 2*t^5 + 2*t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/(741*t^7 - 38*t^6 - 38*t^5 - 38*t^4 - 38*t^3 - 38*t^2 - 38*t + 1).

A165172 Number of reduced words of length n in Coxeter group on 40 generators S_i with relations (S_i)^2 = (S_i S_j)^8 = I.

Original entry on oeis.org

1, 40, 1560, 60840, 2372760, 92537640, 3608967960, 140749750440, 5489240266380, 214080370358400, 8349134442792000, 325616243222649600, 12699033483880036800, 495262305800992828800, 19315229923495904673600
Offset: 0

Views

Author

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

Keywords

Comments

The initial terms coincide with those of A170759, although the two sequences are eventually different.
Computed with MAGMA using commands similar to those used to compute A154638.

Links

Formula

G.f.: (t^8 + 2*t^7 + 2*t^6 + 2*t^5 + 2*t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/(741*t^8 - 38*t^7 - 38*t^6 - 38*t^5 - 38*t^4 - 38*t^3 - 38*t^2 - 38*t + 1).
a(n) = -741*a(n-8) + 38*Sum_{k=1..7} a(n-k). - Wesley Ivan Hurt, Apr 25 2023

A165690 Number of reduced words of length n in Coxeter group on 40 generators S_i with relations (S_i)^2 = (S_i S_j)^9 = I.

Original entry on oeis.org

1, 40, 1560, 60840, 2372760, 92537640, 3608967960, 140749750440, 5489240267160, 214080370418460, 8349134446289520, 325616243404105680, 12699033492713883120, 495262306214038144080, 19315229942277159012720
Offset: 0

Views

Author

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

Keywords

Comments

The initial terms coincide with those of A170759, although the two sequences are eventually different.
Computed with MAGMA using commands similar to those used to compute A154638.

Links

Programs

Formula

G.f. (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)/(741*t^9 - 38*t^8 - 38*t^7 - 38*t^6 - 38*t^5 - 38*t^4 - 38*t^3 -
38*t^2 - 38*t + 1)
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