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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A162365 Number of reduced words of length n in the Weyl group D_23.

Original entry on oeis.org

1, 23, 275, 2277, 14673, 78407, 361514, 1477750, 5461235, 18518565, 58282576, 171815888, 477989151, 1262643305, 3183445871, 7694405993, 17895700206, 40182143330, 87349858045, 184297593435, 378236260170, 756560791350, 1477481301465, 2821499709614, 5276341352226
Offset: 0

Views

Author

John Cannon and N. J. A. Sloane, Dec 01 2009

Keywords

Comments

Differs from A161930 first at index n=23. - R. J. Mathar, Jul 12 2010

References

  • N. Bourbaki, Groupes et alg. de Lie, Chap. 4, 5, 6. (The group is defined in Planche IV.)
  • J. E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge, 1990. See under Poincaré polynomial.

Links

Crossrefs

Row 23 of A162206.
Growth series for groups D_n, n = 3,...,50: A161435, A162207, A162208, A162209, A162210, A162211, A162212, A162248, A162288, A162297, A162300, A162301, A162321, A162327, A162328, A162346, A162347, A162359, A162360, A162364, A162365, A162366, A162367, A162368, A162369, A162370, A162376, A162377, A162378, A162379, A162380, A162381, A162384, A162388, A162389, A162392, A162399, A162402, A162403, A162411, A162412, A162413, A162418, A162452, A162456, A162461, A162469, A162492.

Programs

  • Maple
    # Growth series for D_k, truncated to terms of order M. - N. J. A. Sloane, Aug 07 2021
f := proc(m::integer) (1-x^m)/(1-x) ; end proc:
g := proc(k,M) local a,i; global f;
a:=f(k)*mul(f(2*i),i=1..k-1);
seriestolist(series(a,x,M+1));
end proc;
  • Mathematica
    x = y + y O[y]^(n^2);
(1-x^n) Product[1-x^(2k), {k, 1, n-1}]/(1-x)^n // CoefficientList[#, y]& (* Jean-François Alcover, Mar 25 2020, from A162206 *)

Formula

The growth series for D_k is the polynomial f(k)*Prod_{i=1..k-1} f(2*i), where f(m) = (1-x^m)/(1-x) [Corrected by N. J. A. Sloane, Aug 07 2021]. This is a row of the triangle in A162206.

A162366 Number of reduced words of length n in the Weyl group D_24.

Original entry on oeis.org

1, 24, 299, 2576, 17249, 95656, 457170, 1934920, 7396155, 25914720, 84197296, 256013184, 734002335, 1996645640, 5180091511, 12874497504, 30770197710, 70952341040, 158302199085, 342599792520, 720836052690, 1477396844040, 2954878145505, 5776377855120, 11052719207368
Offset: 0

Views

Author

John Cannon and N. J. A. Sloane, Dec 01 2009

Keywords

Comments

First differs from A161931 at index n=24. - Andrew Howroyd, Mar 17 2025

References

  • N. Bourbaki, Groupes et alg. de Lie, Chap. 4, 5, 6. (The group is defined in Planche IV.)
  • J. E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge, 1990. See under Poincaré polynomial.

Links

Crossrefs

Row 24 of A162206.
Growth series for groups D_n, n = 3,...,50: A161435, A162207, A162208, A162209, A162210, A162211, A162212, A162248, A162288, A162297, A162300, A162301, A162321, A162327, A162328, A162346, A162347, A162359, A162360, A162364, A162365, A162366, A162367, A162368, A162369, A162370, A162376, A162377, A162378, A162379, A162380, A162381, A162384, A162388, A162389, A162392, A162399, A162402, A162403, A162411, A162412, A162413, A162418, A162452, A162456, A162461, A162469, A162492.

Programs

  • Maple
    # Growth series for D_k, truncated to terms of order M. - N. J. A. Sloane, Aug 07 2021
f := proc(m::integer) (1-x^m)/(1-x) ; end proc:
g := proc(k,M) local a,i; global f;
a:=f(k)*mul(f(2*i),i=1..k-1);
seriestolist(series(a,x,M+1));
end proc;
  • Mathematica
    f[m_] := (1-x^m)/(1-x);
With[{k = 24}, CoefficientList[f[k]*Product[f[2i], {i, 1, k-1}] + O[x]^(k-2), x]] (* Jean-François Alcover, Feb 15 2023, after Maple code *)

Formula

The growth series for D_k is the polynomial f(k)*Product_{i=1..k-1} f(2*i), where f(m) = (1-x^m)/(1-x) [Corrected by N. J. A. Sloane, Aug 07 2021]. This is a row of the triangle in A162206.

A162367 Number of reduced words of length n in the Weyl group D_25.

Original entry on oeis.org

1, 25, 324, 2900, 20149, 115805, 572975, 2507895, 9904050, 35818770, 120016066, 376029250, 1110031585, 3106677225, 8286768736, 21161266240, 51931463950, 122883804990, 281186004075, 623785796595, 1344621849285, 2822018693325, 5776896838830, 11553274693950
Offset: 0

Views

Author

John Cannon and N. J. A. Sloane, Dec 01 2009

Keywords

Comments

First differs from A161932 at index n=25. - Andrew Howroyd, Mar 17 2025

References

  • N. Bourbaki, Groupes et alg. de Lie, Chap. 4, 5, 6. (The group is defined in Planche IV.)
  • J. E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge, 1990. See under Poincaré polynomial.

Links

Crossrefs

Row 25 of A162206.
Growth series for groups D_n, n = 3,...,50: A161435, A162207, A162208, A162209, A162210, A162211, A162212, A162248, A162288, A162297, A162300, A162301, A162321, A162327, A162328, A162346, A162347, A162359, A162360, A162364, A162365, A162366, A162367, A162368, A162369, A162370, A162376, A162377, A162378, A162379, A162380, A162381, A162384, A162388, A162389, A162392, A162399, A162402, A162403, A162411, A162412, A162413, A162418, A162452, A162456, A162461, A162469, A162492.

Programs

  • Maple
    # Growth series for D_k, truncated to terms of order M. - N. J. A. Sloane, Aug 07 2021
f := proc(m::integer) (1-x^m)/(1-x) ; end proc:
g := proc(k,M) local a,i; global f;
a:=f(k)*mul(f(2*i),i=1..k-1);
seriestolist(series(a,x,M+1));
end proc;
  • Mathematica
    f[m_] := (1-x^m)/(1-x);
With[{k = 25}, CoefficientList[f[k]*Product[f[2i], {i, 1, k-1}] + O[x]^(k-3), x]] (* Jean-François Alcover, Feb 15 2023, after Maple code *)

Formula

The growth series for D_k is the polynomial f(k)*Product_{i=1..k-1} f(2*i), where f(m) = (1-x^m)/(1-x) [Corrected by N. J. A. Sloane, Aug 07 2021]. This is a row of the triangle in A162206.

A162368 Number of reduced words of length n in the Weyl group D_26.

Original entry on oeis.org

1, 26, 350, 3250, 23399, 139204, 712179, 3220074, 13124124, 48942894, 168958960, 544988210, 1655019795, 4761697020, 13048465756, 34209731996, 86141195946, 209025000936, 490211005011, 1113996801606, 2458618650891, 5280637344216
Offset: 0

Views

Author

John Cannon and N. J. A. Sloane, Dec 01 2009

Keywords

Comments

First differs from A161933 at index n=26. - Andrew Howroyd, Mar 17 2025

References

  • N. Bourbaki, Groupes et alg. de Lie, Chap. 4, 5, 6. (The group is defined in Planche IV.)
  • J. E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge, 1990. See under Poincaré polynomial.

Links

Crossrefs

Row 26 of A162206.
Growth series for groups D_n, n = 3,...,50: A161435, A162207, A162208, A162209, A162210, A162211, A162212, A162248, A162288, A162297, A162300, A162301, A162321, A162327, A162328, A162346, A162347, A162359, A162360, A162364, A162365, A162366, A162367, A162368, A162369, A162370, A162376, A162377, A162378, A162379, A162380, A162381, A162384, A162388, A162389, A162392, A162399, A162402, A162403, A162411, A162412, A162413, A162418, A162452, A162456, A162461, A162469, A162492.

Programs

  • Maple
    # Growth series for D_k, truncated to terms of order M. - N. J. A. Sloane, Aug 07 2021
f := proc(m::integer) (1-x^m)/(1-x) ; end proc:
g := proc(k,M) local a,i; global f;
a:=f(k)*mul(f(2*i),i=1..k-1);
seriestolist(series(a,x,M+1));
end proc;
  • Mathematica
    f[m_] := (1-x^m)/(1-x);
With[{k = 26}, CoefficientList[f[k]*Product[f[2i], {i, 1, k-1}] + O[x]^(k-4), x]] (* Jean-François Alcover, Feb 15 2023, after Maple code *)

Formula

The growth series for D_k is the polynomial f(k)*Product_{i=1..k-1} f(2*i), where f(m) = (1-x^m)/(1-x) [Corrected by N. J. A. Sloane, Aug 07 2021]. This is a row of the triangle in A162206.

A162369 Number of reduced words of length n in the Weyl group D_27.

Original entry on oeis.org

1, 27, 377, 3627, 27026, 166230, 878409, 4098483, 17222607, 66165501, 235124461, 780112671, 2435132466, 7196829486, 20245295242, 54455027238, 140596223184, 349621224120, 839832229131, 1953829030737, 4412447681628, 9693085025844, 20750619208890, 43361428085886
Offset: 0

Views

Author

John Cannon and N. J. A. Sloane, Dec 01 2009

Keywords

References

  • N. Bourbaki, Groupes et alg. de Lie, Chap. 4, 5, 6. (The group is defined in Planche IV.)
  • J. E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge, 1990. See under Poincaré polynomial.

Links

Crossrefs

Row 27 of A162206.
Growth series for groups D_n, n = 3,...,50: A161435, A162207, A162208, A162209, A162210, A162211, A162212, A162248, A162288, A162297, A162300, A162301, A162321, A162327, A162328, A162346, A162347, A162359, A162360, A162364, A162365, A162366, A162367, A162368, A162369, A162370, A162376, A162377, A162378, A162379, A162380, A162381, A162384, A162388, A162389, A162392, A162399, A162402, A162403, A162411, A162412, A162413, A162418, A162452, A162456, A162461, A162469, A162492.

Programs

  • Maple
    # Growth series for D_k, truncated to terms of order M. - N. J. A. Sloane, Aug 07 2021
f := proc(m::integer) (1-x^m)/(1-x) ; end proc:
g := proc(k,M) local a,i; global f;
a:=f(k)*mul(f(2*i),i=1..k-1);
seriestolist(series(a,x,M+1));
end proc;

Formula

The growth series for D_k is the polynomial f(k)*Prod_{i=1..k-1} f(2*i), where f(m) = (1-x^m)/(1-x) [Corrected by N. J. A. Sloane, Aug 07 2021]. This is a row of the triangle in A162206.

A162370 Number of reduced words of length n in the Weyl group D_28.

Original entry on oeis.org

1, 28, 405, 4032, 31058, 197288, 1075697, 5174180, 22396787, 88562288, 323686749, 1103799420, 3538931886, 10735761372, 30981056614, 85436083852, 226032307036, 575653531156, 1415485760287, 3369314791024, 7781762472652, 17474847498496, 38225466707386, 81586894793272
Offset: 0

Views

Author

John Cannon and N. J. A. Sloane, Dec 01 2009

Keywords

References

  • N. Bourbaki, Groupes et alg. de Lie, Chap. 4, 5, 6. (The group is defined in Planche IV.)
  • J. E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge, 1990. See under Poincaré polynomial.

Links

Crossrefs

Row 28 of A162206.
Growth series for groups D_n, n = 3,...,50: A161435, A162207, A162208, A162209, A162210, A162211, A162212, A162248, A162288, A162297, A162300, A162301, A162321, A162327, A162328, A162346, A162347, A162359, A162360, A162364, A162365, A162366, A162367, A162368, A162369, A162370, A162376, A162377, A162378, A162379, A162380, A162381, A162384, A162388, A162389, A162392, A162399, A162402, A162403, A162411, A162412, A162413, A162418, A162452, A162456, A162461, A162469, A162492.

Programs

  • Maple
    # Growth series for D_k, truncated to terms of order M. - N. J. A. Sloane, Aug 07 2021
f := proc(m::integer) (1-x^m)/(1-x) ; end proc:
g := proc(k,M) local a,i; global f;
a:=f(k)*mul(f(2*i),i=1..k-1);
seriestolist(series(a,x,M+1));
end proc;
  • Mathematica
    f[m_] := (1-x^m)/(1-x);
With[{k = 28}, CoefficientList[f[k]*Product[f[2i], {i, 1, k-1}] + O[x]^(k-6), x]] (* Jean-François Alcover, Feb 15 2023, after Maple code *)

Formula

The growth series for D_k is the polynomial f(k)*Product_{i=1..k-1} f(2*i), where f(m) = (1-x^m)/(1-x) [Corrected by N. J. A. Sloane, Aug 07 2021]. This is a row of the triangle in A162206.

A162376 Number of reduced words of length n in the Weyl group D_29.

Original entry on oeis.org

1, 29, 434, 4466, 35524, 232812, 1308509, 6482689, 28879476, 117441764, 441128513, 1544927933, 5083859819, 15819621191, 46800677805, 132236761657, 358269068693, 933922599849, 2349408360136, 5718723151160, 13500485623812, 30975333122308, 69200799829694, 150787694622966
Offset: 0

Views

Author

John Cannon and N. J. A. Sloane, Dec 01 2009

Keywords

References

  • N. Bourbaki, Groupes et alg. de Lie, Chap. 4, 5, 6. (The group is defined in Planche IV.)
  • J. E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge, 1990. See under Poincaré polynomial.

Links

Crossrefs

Row 29 of A162206.
Growth series for groups D_n, n = 3,...,50: A161435, A162207, A162208, A162209, A162210, A162211, A162212, A162248, A162288, A162297, A162300, A162301, A162321, A162327, A162328, A162346, A162347, A162359, A162360, A162364, A162365, A162366, A162367, A162368, A162369, A162370, A162376, A162377, A162378, A162379, A162380, A162381, A162384, A162388, A162389, A162392, A162399, A162402, A162403, A162411, A162412, A162413, A162418, A162452, A162456, A162461, A162469, A162492.

Programs

  • Maple
    # Growth series for D_k, truncated to terms of order M. - N. J. A. Sloane, Aug 07 2021
f := proc(m::integer) (1-x^m)/(1-x) ; end proc:
g := proc(k,M) local a,i; global f;
a:=f(k)*mul(f(2*i),i=1..k-1);
seriestolist(series(a,x,M+1));
end proc;
  • Mathematica
    f[m_] := (1-x^m)/(1-x);
With[{k = 29}, CoefficientList[f[k]*Product[f[2i], {i, 1, k-1}] + O[x]^(k-8), x]] (* Jean-François Alcover, Feb 15 2023, after Maple code *)

Formula

The growth series for D_k is the polynomial f(k)*Product_{i=1..k-1} f(2*i), where f(m) = (1-x^m)/(1-x) [Corrected by N. J. A. Sloane, Aug 07 2021]. This is a row of the triangle in A162206.

A162377 Number of reduced words of length n in the Weyl group D_30.

Original entry on oeis.org

1, 30, 464, 4930, 40454, 273266, 1581775, 8064464, 36943940, 154385704, 595514217, 2140442150, 7224301969, 23043923160, 69844600965, 202081362622, 560350431315, 1494273031164, 3843681391300, 9562404542460, 23062890166272, 54038223288580, 123239023118274, 274026717741240
Offset: 0

Views

Author

John Cannon and N. J. A. Sloane, Dec 01 2009

Keywords

References

  • N. Bourbaki, Groupes et alg. de Lie, Chap. 4, 5, 6. (The group is defined in Planche IV.)
  • J. E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge, 1990. See under Poincaré polynomial.

Links

Crossrefs

Row 30 of A162206.
Growth series for groups D_n, n = 3,...,50: A161435, A162207, A162208, A162209, A162210, A162211, A162212, A162248, A162288, A162297, A162300, A162301, A162321, A162327, A162328, A162346, A162347, A162359, A162360, A162364, A162365, A162366, A162367, A162368, A162369, A162370, A162376, A162377, A162378, A162379, A162380, A162381, A162384, A162388, A162389, A162392, A162399, A162402, A162403, A162411, A162412, A162413, A162418, A162452, A162456, A162461, A162469, A162492.

Programs

  • Maple
    # Growth series for D_k, truncated to terms of order M. - N. J. A. Sloane, Aug 07 2021
f := proc(m::integer) (1-x^m)/(1-x) ; end proc:
g := proc(k,M) local a,i; global f;
a:=f(k)*mul(f(2*i),i=1..k-1);
seriestolist(series(a,x,M+1));
end proc;
  • Mathematica
    f[m_] := (1-x^m)/(1-x);
With[{k = 30}, CoefficientList[f[k]*Product[f[2i], {i, 1, k-1}] + O[x]^(k-9), x]] (* Jean-François Alcover, Feb 15 2023, after Maple code *)

Formula

The growth series for D_k is the polynomial f(k)*Product_{i=1..k-1} f(2*i), where f(m) = (1-x^m)/(1-x) [Corrected by N. J. A. Sloane, Aug 07 2021]. This is a row of the triangle in A162206.

A162378 Number of reduced words of length n in the Weyl group D_31.

Original entry on oeis.org

1, 31, 495, 5425, 45879, 319145, 1900920, 9965384, 46909324, 201295028, 796809245, 2937251395, 10161553364, 33205476524, 103050077489, 305131440111, 865481871426, 2359754902590, 6203436293890, 15765840836350, 38828731002622
Offset: 0

Views

Author

John Cannon and N. J. A. Sloane, Dec 01 2009

Keywords

References

  • N. Bourbaki, Groupes et alg. de Lie, Chap. 4, 5, 6. (The group is defined in Planche IV.)
  • J. E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge, 1990. See under Poincaré polynomial.

Links

Crossrefs

Growth series for groups D_n, n = 3,...,50: A161435, A162207, A162208, A162209, A162210, A162211, A162212, A162248, A162288, A162297, A162300, A162301, A162321, A162327, A162328, A162346, A162347, A162359, A162360, A162364, A162365, A162366, A162367, A162368, A162369, A162370, A162376, A162377, A162378, A162379, A162380, A162381, A162384, A162388, A162389, A162392, A162399, A162402, A162403, A162411, A162412, A162413, A162418, A162452, A162456, A162461, A162469, A162492; also A162206.

Programs

  • Maple
    # Growth series for D_k, truncated to terms of order M. - N. J. A. Sloane, Aug 07 2021
f := proc(m::integer) (1-x^m)/(1-x) ; end proc:
g := proc(k,M) local a,i; global f;
a:=f(k)*mul(f(2*i),i=1..k-1);
seriestolist(series(a,x,M+1));
end proc;
  • Mathematica
    f[m_] := (1-x^m)/(1-x);
With[{k = 31}, CoefficientList[f[k]*Product[f[2i], {i, 1, k-1}] + O[x]^(k-10), x]] (* Jean-François Alcover, Feb 15 2023, after Maple code *)

Formula

The growth series for D_k is the polynomial f(k)*Product_{i=1..k-1} f(2*i), where f(m) = (1-x^m)/(1-x) [Corrected by N. J. A. Sloane, Aug 07 2021]. This is a row of the triangle in A162206.

A162379 Number of reduced words of length n in the Weyl group D_32.

Original entry on oeis.org

1, 32, 527, 5952, 51831, 370976, 2271896, 12237280, 59146604, 260441632, 1057250877, 3994502272, 14156055636, 47361532160, 150411609649, 455543049760, 1321024921186, 3680779823776, 9884216117666, 25650056954016
Offset: 0

Views

Author

John Cannon and N. J. A. Sloane, Dec 01 2009

Keywords

References

  • N. Bourbaki, Groupes et alg. de Lie, Chap. 4, 5, 6. (The group is defined in Planche IV.)
  • J. E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge, 1990. See under Poincaré polynomial.

Links

Crossrefs

Growth series for groups D_n, n = 3,...,50: A161435, A162207, A162208, A162209, A162210, A162211, A162212, A162248, A162288, A162297, A162300, A162301, A162321, A162327, A162328, A162346, A162347, A162359, A162360, A162364, A162365, A162366, A162367, A162368, A162369, A162370, A162376, A162377, A162378, A162379, A162380, A162381, A162384, A162388, A162389, A162392, A162399, A162402, A162403, A162411, A162412, A162413, A162418, A162452, A162456, A162461, A162469, A162492; also A162206.

Programs

  • Maple
    # Growth series for D_k, truncated to terms of order M. - N. J. A. Sloane, Aug 07 2021
f := proc(m::integer) (1-x^m)/(1-x) ; end proc:
g := proc(k,M) local a,i; global f;
a:=f(k)*mul(f(2*i),i=1..k-1);
seriestolist(series(a,x,M+1));
end proc;
  • Mathematica
    f[m_] := (1-x^m)/(1-x);
With[{k = 32}, CoefficientList[f[k]*Product[f[2i], {i, 1, k-1}] + O[x]^(k-12), x]] (* Jean-François Alcover, Feb 15 2023, after Maple code *)

Formula

The growth series for D_k is the polynomial f(k)*Product_{i=1..k-1} f(2*i), where f(m) = (1-x^m)/(1-x) [Corrected by N. J. A. Sloane, Aug 07 2021]. This is a row of the triangle in A162206.
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