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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A167370 Number of reduced words of length n in Coxeter group on 30 generators S_i with relations (S_i)^2 = (S_i S_j)^14 = I.

Original entry on oeis.org

1, 30, 870, 25230, 731670, 21218430, 615334470, 17844699630, 517496289270, 15007392388830, 435214379276070, 12621216999006030, 366015292971174870, 10614443496164071230, 307818861388758065235, 8926746980273983879200, 258875662427945532131400
Offset: 0

Views

Author

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

Keywords

Comments

The initial terms coincide with those of A170749, although the two sequences are eventually different.
Computed with MAGMA using commands similar to those used to compute A154638.

Links

Crossrefs

Cf. A154638, A170749.

Programs

  • Mathematica
    CoefficientList[Series[(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)/ (406*t^14 - 28*t^13 - 28*t^12 - 28*t^11 - 28*t^10 - 28*t^9 - 28*t^8 - 28*t^7 - 28*t^6 - 28*t^5 - 28*t^4 - 28*t^3 - 28*t^2 - 28*t + 1), {t, 0, 50}], t] (* G. C. Greubel, Jun 11 2016 *)
coxG[{14,406,-28}] (* The coxG program is at A169452 *) (* Harvey P. Dale, Jul 04 2017 *)

Formula

G.f.: (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)/(406*t^14 - 28*t^13 - 28*t^12 - 28*t^11 - 28*t^10 - 28*t^9 - 28*t^8 - 28*t^7 - 28*t^6 - 28*t^5 - 28*t^4 - 28*t^3 - 28*t^2 - 28*t + 1).

A167715 Number of reduced words of length n in Coxeter group on 30 generators S_i with relations (S_i)^2 = (S_i S_j)^15 = I.

Original entry on oeis.org

1, 30, 870, 25230, 731670, 21218430, 615334470, 17844699630, 517496289270, 15007392388830, 435214379276070, 12621216999006030, 366015292971174870, 10614443496164071230, 307818861388758065670, 8926746980273983903995
Offset: 0

Views

Author

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

Keywords

Comments

The initial terms coincide with those of A170749, although the two sequences are eventually different.
Computed with MAGMA using commands similar to those used to compute A154638.

Links

Programs

  • Mathematica
    CoefficientList[Series[(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)/(406*t^15 - 28*t^14 - 28*t^13 - 28*t^12 - 28*t^11 - 28*t^10 - 28*t^9 - 28*t^8 - 28*t^7 - 28*t^6 - 28*t^5 - 28*t^4 - 28*t^3 - 28*t^2 - 28*t + 1), {t, 0, 50}], t] (* G. C. Greubel, Jun 20 2016 *)

Formula

G.f.: (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)/(406*t^15 - 28*t^14 - 28*t^13 - 28*t^12 - 28*t^11 - 28*t^10 - 28*t^9 - 28*t^8 - 28*t^7 - 28*t^6 - 28*t^5 - 28*t^4 - 28*t^3 - 28*t^2 - 28*t + 1).

A167945 Number of reduced words of length n in Coxeter group on 30 generators S_i with relations (S_i)^2 = (S_i S_j)^16 = I.

Original entry on oeis.org

1, 30, 870, 25230, 731670, 21218430, 615334470, 17844699630, 517496289270, 15007392388830, 435214379276070, 12621216999006030, 366015292971174870, 10614443496164071230, 307818861388758065670, 8926746980273983904430
Offset: 0

Views

Author

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

Keywords

Comments

The initial terms coincide with those of A170749, although the two sequences are eventually different.
Computed with MAGMA using commands similar to those used to compute A154638.

Links

Crossrefs

Cf. A154638, A169452, A170749.

Programs

  • Magma
    R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1+x)*(1-x^16)/(1-29*x+434*x^16-406*x^17) )); // G. C. Greubel, Sep 07 2023
  • Mathematica
    CoefficientList[Series[(1+t)*(1-t^16)/(1-29*t+434*t^16-406*t^17), {t, 0, 50}], t] (* G. C. Greubel, Jul 02 2016; Sep 07 2023 *)
coxG[{16,406,-28}] (* The coxG program is at A169452 *) (* Harvey P. Dale, Dec 15 2017 *)
  • SageMath
    def A167945_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1+x)*(1-x^16)/(1-29*x+434*x^16-406*x^17) ).list()
A167945_list(40) # G. C. Greubel, Sep 07 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)/( 406*t^16 - 28*t^15 - 28*t^14 - 28*t^13 - 28*t^12 - 28*t^11 - 28*t^10 - 28*t^9 - 28*t^8 - 28*t^7 - 28*t^6 - 28*t^5 - 28*t^4 - 28*t^3 - 28*t^2 - 28*t + 1).
From G. C. Greubel, Sep 07 2023: (Start)
G.f.: (1+t)*(1-t^16)/(1 - 29*t + 434*t^16 - 406*t^17).
a(n) = 28*Sum_{j=1..15} a(n-j) - 406*a(n-16). (End)

A168707 Number of reduced words of length n in Coxeter group on 30 generators S_i with relations (S_i)^2 = (S_i S_j)^17 = I.

Original entry on oeis.org

1, 30, 870, 25230, 731670, 21218430, 615334470, 17844699630, 517496289270, 15007392388830, 435214379276070, 12621216999006030, 366015292971174870, 10614443496164071230, 307818861388758065670, 8926746980273983904430
Offset: 0

Views

Author

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

Keywords

Comments

The initial terms coincide with those of A170749, although the two sequences are eventually different.
First disagreement at index 17: a(17) = 7507394210410420463625195, A170749(17) = 7507394210410420463625630. - Klaus Brockhaus, Mar 28 2011
Computed with MAGMA using commands similar to those used to compute A154638.

Links

Crossrefs

Cf. A170749 (G.f.: (1+x)/(1-29*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)/(406*t^17 - 28*t^16 - 28*t^15 - 28*t^14 - 28*t^13 - 28*t^12 - 28*t^11 - 28*t^10 - 28*t^9 - 28*t^8 - 28*t^7 - 28*t^6 - 28*t^5 - 28*t^4 - 28*t^3 - 28*t^2 - 28*t + 1), {t,0,50}], t] (* G. C. Greubel, Aug 04 2016 *)
coxG[{17,46,-28}] (* The coxG program is at A169452 *) (* Harvey P. Dale, Sep 09 2019 *)

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)/(406*t^17 - 28*t^16 - 28*t^15 - 28*t^14 - 28*t^13 - 28*t^12 - 28*t^11 - 28*t^10 - 28*t^9 - 28*t^8 - 28*t^7 - 28*t^6 - 28*t^5 - 28*t^4 - 28*t^3 - 28*t^2 - 28*t + 1).

A168755 Number of reduced words of length n in Coxeter group on 30 generators S_i with relations (S_i)^2 = (S_i S_j)^18 = I.

Original entry on oeis.org

1, 30, 870, 25230, 731670, 21218430, 615334470, 17844699630, 517496289270, 15007392388830, 435214379276070, 12621216999006030, 366015292971174870, 10614443496164071230, 307818861388758065670, 8926746980273983904430
Offset: 0

Views

Author

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

Keywords

Comments

The initial terms coincide with those of A170749, although the two sequences are eventually different.
First disagreement at index 18: a(18) = 217714432101902193445142835, A170749(18) = 217714432101902193445143270. - Klaus Brockhaus, Mar 26 2011
Computed with MAGMA using commands similar to those used to compute A154638.

Links

Crossrefs

Cf. A170749 (G.f.: (1+x)/(1-29*x)).

Programs

  • Mathematica
    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)/(406*t^18 - 28*t^17 - 28*t^16 - 28*t^15 - 28*t^14 - 28*t^13 - 28*t^12 - 28*t^11 - 28*t^10 - 28*t^9 - 28*t^8 - 28*t^7 - 28*t^6 - 28*t^5 - 28*t^4 - 28*t^3 - 28*t^2 - 28*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)/(406*t^18 - 28*t^17 - 28*t^16 - 28*t^15 - 28*t^14 - 28*t^13 - 28*t^12 - 28*t^11 - 28*t^10 - 28*t^9 - 28*t^8 - 28*t^7 - 28*t^6 - 28*t^5 - 28*t^4 - 28*t^3 - 28*t^2 - 28*t + 1).

A168803 Number of reduced words of length n in Coxeter group on 30 generators S_i with relations (S_i)^2 = (S_i S_j)^19 = I.

Original entry on oeis.org

1, 30, 870, 25230, 731670, 21218430, 615334470, 17844699630, 517496289270, 15007392388830, 435214379276070, 12621216999006030, 366015292971174870, 10614443496164071230, 307818861388758065670, 8926746980273983904430
Offset: 0

Views

Author

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

Keywords

Comments

The initial terms coincide with those of A170749, although the two sequences are eventually different.
First disagreement at index 19: a(19) = 6313718530955163609909154395, A170749(19) = 6313718530955163609909154830. - Klaus Brockhaus, Apr 01 2011
Computed with MAGMA using commands similar to those used to compute A154638.

Links

Crossrefs

Cf. A170749 (G.f.: (1+x)/(1-29*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)/(406*t^19 - 28*t^18 - 28*t^17 - 28*t^16 - 28*t^15 - 28*t^14 - 28*t^13 - 28*t^12 - 28*t^11 - 28*t^10 - 28*t^9 - 28*t^8 - 28*t^7 - 28*t^6 - 28*t^5 - 28*t^4 - 28*t^3 - 28*t^2 - 28*t + 1), {t, 0, 500}], t] (* G. C. Greubel, Aug 16 2016 *)
coxG[{19,406,-28}] (* The coxG program is at A169452 *) (* Harvey P. Dale, Dec 11 2016 *)

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)/(406*t^19 - 28*t^18 - 28*t^17 - 28*t^16 - 28*t^15 - 28*t^14 - 28*t^13 - 28*t^12 - 28*t^11 - 28*t^10 - 28*t^9 - 28*t^8 - 28*t^7 - 28*t^6 - 28*t^5 - 28*t^4 - 28*t^3 - 28*t^2 - 28*t + 1).

A169091 Number of reduced words of length n in Coxeter group on 30 generators S_i with relations (S_i)^2 = (S_i S_j)^25 = I.

Original entry on oeis.org

1, 30, 870, 25230, 731670, 21218430, 615334470, 17844699630, 517496289270, 15007392388830, 435214379276070, 12621216999006030, 366015292971174870, 10614443496164071230, 307818861388758065670, 8926746980273983904430
Offset: 0

Views

Author

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

Keywords

Comments

The initial terms coincide with those of A170749, although the two sequences are eventually different.
First disagreement at index 25: a(25) = 3755547024441991720544511984283789995, A170749(25) = 3755547024441991720544511984283790430. - Klaus Brockhaus, Apr 25 2011
Computed with MAGMA using commands similar to those used to compute A154638.

Links

Crossrefs

Cf. A170749 (G.f.: (1+x)/(1-29*x)).

Programs

Formula

G.f.: (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)/(406*t^25 - 28*t^24 - 28*t^23 - 28*t^22 - 28*t^21 - 28*t^20 - 28*t^19 - 28*t^18 - 28*t^17 - 28*t^16 - 28*t^15 - 28*t^14 - 28*t^13 - 28*t^12 - 28*t^11 - 28*t^10 - 28*t^9 - 28*t^8 - 28*t^7 - 28*t^6 - 28*t^5 - 28*t^4 - 28*t^3 - 28*t^2 - 28*t + 1).

A170183 Number of reduced words of length n in Coxeter group on 30 generators S_i with relations (S_i)^2 = (S_i S_j)^39 = I.

Original entry on oeis.org

1, 30, 870, 25230, 731670, 21218430, 615334470, 17844699630, 517496289270, 15007392388830, 435214379276070, 12621216999006030, 366015292971174870, 10614443496164071230, 307818861388758065670, 8926746980273983904430
Offset: 0

Views

Author

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

Keywords

Comments

The initial terms coincide with those of A170749, although the two sequences are eventually different.
First disagreement is at index 39, the difference is 435. - Vincenzo Librandi, Dec 10 2009
Computed with MAGMA using commands similar to those used to compute A154638.

Links

Crossrefs

Cf. A170749.

Programs

  • Magma
    /* Alternatively */ m:=16; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((t^40+t^39-t-1)/(406*t^40-434*t^39+29*t-1)));  // Bruno Berselli, Sep 20 2011
  • Mathematica
    With[{num=Total[2t^Range[38]]+t^39+1,den=Total[-28 t^Range[38]]+ 406t^39+1}, CoefficientList[Series[num/den,{t,0,30}],t]] (* Harvey P. Dale, Sep 20 2011 *)

Formula

G.f.: (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) /(406*t^39 - 28*t^38 - 28*t^37 - 28*t^36 - 28*t^35 - 28*t^34 - 28*t^33 - 28*t^32 - 28*t^31 - 28*t^30 - 28*t^29 - 28*t^28 - 28*t^27 - 28*t^26 - 28*t^25 - 28*t^24 - 28*t^23 - 28*t^22 - 28*t^21 - 28*t^20 - 28*t^19 - 28*t^18 - 28*t^17 - 28*t^16 - 28*t^15 - 28*t^14 - 28*t^13 - 28*t^12 - 28*t^11 - 28*t^10 - 28*t^9 - 28*t^8 - 28*t^7 - 28*t^6 - 28*t^5 - 28*t^4 - 28*t^3 - 28*t^2 - 28*t + 1)

A170711 Number of reduced words of length n in Coxeter group on 30 generators S_i with relations (S_i)^2 = (S_i S_j)^50 = I.

Original entry on oeis.org

1, 30, 870, 25230, 731670, 21218430, 615334470, 17844699630, 517496289270, 15007392388830, 435214379276070, 12621216999006030, 366015292971174870, 10614443496164071230, 307818861388758065670, 8926746980273983904430
Offset: 0

Views

Author

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

Keywords

Comments

The initial terms coincide with those of A170749, 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 435. - Vincenzo Librandi, Dec 06 2012

Links

Programs

  • Mathematica
    With[{num=Total[2 t^Range[49]] + t^50 + 1, den = Total[-28t^Range[49]] + 406 t^50 + 1}, CoefficientList[Series[num/den, {t, 0, 30}], t]] (* Vincenzo Librandi, Dec 06 2012 *)
coxG[{50,406,-28}] (* The coxG program is at A169452 *) (* Harvey P. Dale, Dec 02 2017 *)

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)/(406*t^50 - 28*t^49 - 28*t^48 - 28*t^47 - 28*t^46 - 28*t^45 -
28*t^44 - 28*t^43 - 28*t^42 - 28*t^41 - 28*t^40 - 28*t^39 - 28*t^38 -
28*t^37 - 28*t^36 - 28*t^35 - 28*t^34 - 28*t^33 - 28*t^32 - 28*t^31 -
28*t^30 - 28*t^29 - 28*t^28 - 28*t^27 - 28*t^26 - 28*t^25 - 28*t^24 -
28*t^23 - 28*t^22 - 28*t^21 - 28*t^20 - 28*t^19 - 28*t^18 - 28*t^17 -
28*t^16 - 28*t^15 - 28*t^14 - 28*t^13 - 28*t^12 - 28*t^11 - 28*t^10 -
28*t^9 - 28*t^8 - 28*t^7 - 28*t^6 - 28*t^5 - 28*t^4 - 28*t^3 - 28*t^2 -
28*t + 1).

A162833 Number of reduced words of length n in Coxeter group on 30 generators S_i with relations (S_i)^2 = (S_i S_j)^3 = I.

Original entry on oeis.org

1, 30, 870, 24795, 706440, 20121360, 573111630, 16323709080, 464941707720, 13242748348620, 377187895691040, 10743291699776160, 305996872843541880, 8715586321562342880, 248242553013255652320, 7070593171000425862320
Offset: 0

Views

Author

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

Keywords

Comments

The initial terms coincide with those of A170749, 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^3 + 2*t^2 + 2*t + 1)/(406*t^3 - 28*t^2 - 28*t + 1)
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