A266773 Molien series for invariants of finite Coxeter group D_10 (bisected).

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

Author

N. J. A. Sloane, Jan 11 2016

Keywords

• nonn

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.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for Molien series

Crossrefs

Molien series for finite Coxeter groups D_3 through D_12 are A266755, A266769, A266768, A003402, and A266770-A266775.

Programs

Magma

R:=PowerSeriesRing(Integers(), 50); Coefficients(R!( 1/((1-x^5)*(&*[1-x^j: j in [1..9]])) )); // 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 *)

PARI

Vec(1/((1-x^5)*prod(j=1,9,1-x^j)) +O('x^50)) \\ G. C. Greubel, Feb 03 2020

Sage

def A266773_list(prec):
    P. = 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

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