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.

Previous Showing 11-20 of 49 results. Next

A162346 Number of reduced words of length n in the Weyl group D_18.

Original entry on oeis.org

1, 18, 170, 1122, 5813, 25176, 94791, 318630, 974643, 2752112, 7253764, 18003544, 42378246, 95162260, 204856291, 424515042, 849825768, 1648470894, 3106669574, 5701318526, 10209535012, 17871859722, 30631153147, 51476598044, 84931517948, 137735283228, 219783774729
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 18 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.

A162347 Number of reduced words of length n in the Weyl group D_19.

Original entry on oeis.org

1, 19, 189, 1311, 7124, 32300, 127091, 445721, 1420364, 4172476, 11426240, 29429784, 71808030, 166970290, 371826581, 796341623, 1646167391, 3294638285, 6401307860, 12102626403, 22312161567, 40184022241, 70815180079, 122291797486, 207223385049, 344958892116
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 19 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.

A162359 Number of reduced words of length n in the Weyl group D_20.

Original entry on oeis.org

1, 20, 209, 1520, 8644, 40944, 168035, 613756, 2034120, 6206596, 17632836, 47062620, 118870650, 285840940, 657667521, 1454009144, 3100176535, 6394814820, 12796122680, 24898749084, 47210910669, 87394933080, 158210114281, 280501917580, 487725327805, 832684314712
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 20 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.

A162207 Number of reduced words of length n in the Weyl group D_4.

Original entry on oeis.org

1, 4, 9, 16, 23, 28, 30, 28, 23, 16, 9, 4, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
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
    n = 4;
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.

A162208 Number of reduced words of length n in the Weyl group D_5.

Original entry on oeis.org

1, 5, 14, 30, 54, 85, 120, 155, 185, 205, 212, 205, 185, 155, 120, 85, 54, 30, 14, 5, 1
Offset: 0

Views

Author

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

Keywords

References

  • N. Bourbaki, Groupes et algèbres 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 5 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
    A162208g := proc(m::integer)
    (1-x^m)/(1-x) ;
end proc:
A162208 := proc(n,k)
    g := A162208g(k);
    for m from 2 to 2*k-2 by 2 do
        g := g*A162208g(m) ;
    end do:
    g := expand(g) ;
    coeftayl(g,x=0,n) ;
end proc:
seq( A162208(n,5),n=0..60) ; # R. J. Mathar, Jan 19 2016
  • Mathematica
    n = 5;
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.

A162209 Number of reduced words of length n in the Weyl group D_6.

Original entry on oeis.org

1, 6, 20, 50, 104, 190, 314, 478, 679, 908, 1151, 1390, 1605, 1776, 1886, 1924, 1886, 1776, 1605, 1390, 1151, 908, 679, 478, 314, 190, 104, 50, 20, 6, 1
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 6 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
    n = 6;
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.

A162210 Number of reduced words of length n in the Weyl group D_7.

Original entry on oeis.org

1, 7, 27, 77, 181, 371, 686, 1169, 1862, 2800, 4005, 5481, 7210, 9149, 11230, 13363, 15442, 17353, 18983, 20230, 21013, 21280, 21013, 20230, 18983, 17353, 15442, 13363, 11230, 9149, 7210, 5481, 4005, 2800, 1862, 1169, 686, 371, 181, 77, 27, 7, 1
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 7 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
    n = 7;
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.

A162211 Number of reduced words of length n in the Weyl group D_8.

Original entry on oeis.org

1, 8, 35, 112, 293, 664, 1350, 2520, 4388, 7208, 11263, 16848, 24248, 33712, 45425, 59480, 75853, 94384, 114766, 136544, 159125, 181800, 203777, 224224, 242318, 257296, 268504, 275440, 277788, 275440, 268504, 257296, 242318, 224224, 203777, 181800, 159125
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 8 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
    n = 8;
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.

A162360 Number of reduced words of length n in the Weyl group D_21.

Original entry on oeis.org

1, 21, 230, 1750, 10394, 51338, 219373, 833129, 2867249, 9073845, 26706681, 73769301, 192639951, 478480891, 1136148412, 2590157556, 5690334091, 12085148911, 24881271591, 49780020675, 96990931345, 184385864444, 342595978914, 623097897805, 1110823232734
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 21 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 = 21}, CoefficientList[f[k]*Product[f[2i], {i, 1, k-1}] + O[x]^(k+1), 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.

A162364 Number of reduced words of length n in the Weyl group D_22.

Original entry on oeis.org

1, 22, 252, 2002, 12396, 63734, 283107, 1116236, 3983485, 13057330, 39764011, 113533312, 306173263, 784654154, 1920802566, 4510960122, 10201294213, 22286443124, 47167714715, 96947735390, 193938666735, 378324531180, 720920510114, 1344018408128, 2454841642382
Offset: 0

Views

Author

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

Keywords

Comments

First differs from A161900 at index n=22. - 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 22 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 = 22}, CoefficientList[f[k]*Product[f[2i], {i, 1, k-1}] + O[x]^k, 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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