OFFSET
0,2
REFERENCES
N. Bourbaki, Groupes et Algèbres de Lie, Chap. 4, 5 and 6, Hermann, Paris, 1968. See Chap. VI, Section 4, Problem 10a, page 231, W(t).
J. E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge, 1990. See under Poincaré polynomial.
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.
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 *)
x =.; n = 11; CoefficientList[ Product[1 - x^(2 k), {k, 1, n - 1}] (1 - x^n) /(1 - x)^n // Expand, x] (* Michael Somos, Aug 06 2021 *)
CROSSREFS
The growth series for D_k, k >= 3, are also the rows of the triangle 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; also A162206.
KEYWORD
nonn
AUTHOR
John Cannon and N. J. A. Sloane, Dec 01 2009
EXTENSIONS
Entry revised by N. J. A. Sloane, Jan 17 2016
STATUS
approved