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A266773
Molien series for invariants of finite Coxeter group D_10 (bisected).
1
1, 1, 2, 3, 5, 8, 12, 17, 25, 35, 49, 66, 90, 119, 158, 206, 267, 342, 437, 551, 694, 865, 1074, 1324, 1627, 1985, 2414, 2919, 3518, 4219, 5045, 6003, 7125, 8422, 9927, 11660, 13660, 15949, 18578, 21575, 24998, 28884, 33303, 38298, 43955, 50329, 57513, 65581, 74645, 84786
OFFSET
0,3
COMMENTS
The Molien series for the finite Coxeter group of type D_k (k >= 3) has G.f. = 1/Prod_i (1-x^(1+m_i)) where the m_i are [1,3,5,...,2k-3,k-1]. If k is even only even powers of x appear, and we bisect the sequence.
REFERENCES
J. E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge, 1990. See Table 3.1, page 59.
FORMULA
G.f.: 1/((1-t^2)*(1-t^4)*(1-t^6)*(1-t^8)*(1-t^10)^2*(1-t^12)*(1-t^14)*(1-t^16)*(1-t^18)), bisected.
G.f.: 1/( (1-x^5)*(Product_{j=1..9} 1-x^j) ). - G. C. Greubel, Feb 03 2020
MAPLE
seq(coeff(series(1/((1-x^5)*mul(1-x^j, j=1..9)), x, n+1), x, n), n = 0..50); # G. C. Greubel, Feb 03 2020
MATHEMATICA
CoefficientList[Series[1/((1-x^5)*Product[1-x^j, {j, 9}]), {x, 0, 50}], x] (* G. C. Greubel, Feb 03 2020 *)
PROG
(PARI) Vec(1/((1-x^5)*prod(j=1, 9, 1-x^j)) +O('x^50)) \\ G. C. Greubel, Feb 03 2020
(Magma) R<x>:=PowerSeriesRing(Integers(), 50); Coefficients(R!( 1/((1-x^5)*(&*[1-x^j: j in [1..9]])) )); // G. C. Greubel, Feb 03 2020
(Sage)
def A266773_list(prec):
P.<x> = PowerSeriesRing(ZZ, prec)
return P( 1/((1-x^5)*product(1-x^j for j in (1..9))) ).list()
A266773_list(50) # G. C. Greubel, Feb 03 2020
CROSSREFS
Molien series for finite Coxeter groups D_3 through D_12 are A266755, A266769, A266768, A003402, and A266770-A266775.
Sequence in context: A129504 A241553 A241549 * A347448 A024789 A318028
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Jan 11 2016
STATUS
approved