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A162208
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Number of reduced words of length n in the Weyl group D_5.
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31
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1, 5, 14, 30, 54, 85, 120, 155, 185, 205, 212, 205, 185, 155, 120, 85, 54, 30, 14, 5, 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
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listen;
history;
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internal format)
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OFFSET
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0,2
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REFERENCES
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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.
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LINKS
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Table of n, a(n) for n=0..91.
Index entries for growth series for groups
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FORMULA
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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.
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MAPLE
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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
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MATHEMATICA
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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 *)
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CROSSREFS
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The growth series for D_k, k >= 3, are also the rows of the triangle A162206.
Growth series for groups D_n, n = 3,...,32: 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; also A162206
Sequence in context: A076042 A231669 A256986 * A161698 A049791 A053461
Adjacent sequences: A162205 A162206 A162207 * A162209 A162210 A162211
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KEYWORD
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nonn
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AUTHOR
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John Cannon and N. J. A. Sloane, Dec 01 2009
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STATUS
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approved
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