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Growth series for groups D_n, n = 3,...,3250: 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.
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The growth series for D_k is the polynomial f(k)*Prod_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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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 *)
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Computed with MAGMA using commands similar to those used to compute A161409.
<a href="/index/Gre#GROWTH">Index entries for growth series for groups</a>
G.f. The growth series for D_m k is the polynomial f(nk) * Product( f(2i), Prod_{i=1..nk-1 )/ } f(12*i)^n, , where f(km) = (1-x^km)/(1-x) [Corrected by _N. J. A. Only finitely many terms are nonzeroSloane_, Aug 07 2021]. This is a row of the triangle in A162206.
# 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;
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
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J. E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge, 1990. See under Poincare Poincaré polynomial.
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