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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A162380 Number of reduced words of length n in the Weyl group D_33.

Original entry on oeis.org

1, 33, 560, 6512, 58343, 429319, 2701215, 14938495, 74085099, 334526731, 1391777608, 5386279880, 19542335516, 66903867676, 217315477325, 672858527085, 1993883448271, 5674663272047, 15558879389713, 41208936343729
Offset: 0

Views

Author

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

Keywords

Comments

Computed with MAGMA using commands similar to those used to compute A161409.

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.

Formula

G.f. for D_m is the polynomial f(n) * Product( f(2i), i=1..n-1 )/ f(1)^n, where f(k) = 1-x^k. Only finitely many terms are nonzero. This is a row of the triangle in A162206.

A162381 Number of reduced words of length n in the Weyl group D_34.

Original entry on oeis.org

1, 34, 594, 7106, 65449, 494768, 3195983, 18134478, 92219577, 426746308, 1818523916, 7204803796, 26747139312, 93651006988, 310966484313, 983825011398, 2977708459669, 8652371731716, 24211251121429, 65420187465158
Offset: 0

Views

Author

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

Keywords

Comments

Computed with MAGMA using commands similar to those used to compute A161409.

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.

Formula

G.f. for D_m is the polynomial f(n) * Product( f(2i), i=1..n-1 )/ f(1)^n, where f(k) = 1-x^k. Only finitely many terms are nonzero. This is a row of the triangle in A162206.

A162384 Number of reduced words of length n in the Weyl group D_35.

Original entry on oeis.org

1, 35, 629, 7735, 73184, 567952, 3763935, 21898413, 114117990, 540864298, 2359388214, 9564192010, 36311331322, 129962338310, 440928822623, 1424753834021, 4402462293690, 13054834025406, 37266085146835
Offset: 0

Views

Author

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

Keywords

Comments

Computed with MAGMA using commands similar to those used to compute A161409.

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.

Formula

G.f. for D_m is the polynomial f(n) * Product( f(2i), i=1..n-1 )/ f(1)^n, where f(k) = 1-x^k. Only finitely many terms are nonzero. This is a row of the triangle in A162206.

A162388 Number of reduced words of length n in the Weyl group D_36.

Original entry on oeis.org

1, 36, 665, 8400, 81584, 649536, 4413471, 26311884, 140429874, 681294172, 3040682386, 12604874396, 48916205718, 178878544028, 619807366651, 2044561200672, 6447023494362, 19501857519768, 56767942666603
Offset: 0

Views

Author

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

Keywords

Comments

Computed with MAGMA using commands similar to those used to compute A161409.

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.

Formula

G.f. for D_m is the polynomial f(n) * Product( f(2i), i=1..n-1 )/ f(1)^n, where f(k) = 1-x^k. Only finitely many terms are nonzero. This is a row of the triangle in A162206.

A162389 Number of reduced words of length n in the Weyl group D_37.

Original entry on oeis.org

1, 37, 702, 9102, 90686, 740222, 5153693, 31465577, 171895451, 853189623, 3893872009, 16498746405, 65414952123, 244293496151, 864100862802, 2908662063474, 9355685557836, 28857543077604, 85625485744207
Offset: 0

Views

Author

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

Keywords

Comments

Computed with MAGMA using commands similar to those used to compute A161409.

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.

Formula

G.f. for D_m is the polynomial f(n) * Product( f(2i), i=1..n-1 )/ f(1)^n, where f(k) = 1-x^k. Only finitely many terms are nonzero. This is a row of the triangle in A162206.

A162392 Number of reduced words of length n in the Weyl group D_38.

Original entry on oeis.org

1, 38, 740, 9842, 100528, 840750, 5994443, 37460020, 209355471, 1062545094, 4956417103, 21455163508, 86870115631, 331163611782, 1195264474584, 4103926538058, 13459612095894, 42317155173498, 127942640917705
Offset: 0

Views

Author

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

Keywords

Comments

Computed with MAGMA using commands similar to those used to compute A161409.

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.

Formula

G.f. for D_m is the polynomial f(n) * Product( f(2i), i=1..n-1 )/ f(1)^n, where f(k) = 1-x^k. Only finitely many terms are nonzero. This is a row of the triangle in A162206.

A162399 Number of reduced words of length n in the Weyl group D_39.

Original entry on oeis.org

1, 39, 779, 10621, 111149, 951899, 6946342, 44406362, 253761833, 1316306927, 6272724030, 27727887538, 114598003169, 445761614951, 1641026089535, 5744952627593, 19204564723487, 61521719896985, 189464360814690
Offset: 0

Views

Author

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

Keywords

Comments

Computed with MAGMA using commands similar to those used to compute A161409.

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.

Formula

G.f. for D_m is the polynomial f(n) * Product( f(2i), i=1..n-1 )/ f(1)^n, where f(k) = 1-x^k. Only finitely many terms are nonzero. This is a row of the triangle in A162206.

A162402 Number of reduced words of length n in the Weyl group D_40.

Original entry on oeis.org

1, 40, 819, 11440, 122589, 1074488, 8020830, 52427192, 306189025, 1622495952, 7895219982, 35623107520, 150221110689, 595982725640, 2237008815175, 7981961442768, 27186526166255, 88708246063240, 278172606877930
Offset: 0

Views

Author

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

Keywords

Comments

Computed with MAGMA using commands similar to those used to compute A161409.

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.

Formula

G.f. for D_m is the polynomial f(n) * Product( f(2i), i=1..n-1 )/ f(1)^n, where f(k) = 1-x^k. Only finitely many terms are nonzero. This is a row of the triangle in A162206.

A162403 Number of reduced words of length n in the Weyl group D_41.

Original entry on oeis.org

1, 41, 860, 12300, 134889, 1209377, 9230207, 61657399, 367846424, 1990342376, 9885562358, 45508669878, 195729780567, 791712506207, 3028721321382, 11010682764150, 38197208930405, 126905454993645, 405078061871575
Offset: 0

Views

Author

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

Keywords

Comments

Computed with MAGMA using commands similar to those used to compute A161409.

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

Cf. A161409.
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
    f:= k -> 1-x^k:
g:= n -> f(n)*mul(f(2*i),i=1..n-1)/f(1)^n:
S:= expand(normal(g(41))):
seq(coeff(S,x,j),j=0..degree(S,x)); # Robert Israel, Oct 07 2015
  • Mathematica
    n = 41;
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

G.f. for D_m is the polynomial f(n) * Product( f(2i), i=1..n-1 )/ f(1)^n, where f(k) = 1-x^k. Only finitely many terms are nonzero. This is a row of the triangle in A162206.

A162411 Number of reduced words of length n in the Weyl group D_42.

Original entry on oeis.org

1, 42, 902, 13202, 148091, 1357468, 10587675, 72245074, 440091498, 2430433874, 12315996232, 57824666110, 253554446677, 1045266952884, 4073988274266, 15084671038416, 53281879968821, 180187334962466, 585265396834041
Offset: 0

Views

Author

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

Keywords

Comments

Computed with MAGMA using commands similar to those used to compute A161409.

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.

Formula

G.f. for D_m is the polynomial f(n) * Product( f(2i), i=1..n-1 )/ f(1)^n, where f(k) = 1-x^k. Only finitely many terms are nonzero. This is a row of the triangle in A162206.
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