A266772 Molien series for invariants of finite Coxeter group D_9.

1, 0, 1, 0, 2, 0, 3, 0, 5, 1, 7, 1, 11, 2, 15, 3, 22, 5, 30, 7, 41, 11, 54, 15, 73, 22, 94, 30, 123, 41, 157, 54, 201, 73, 252, 94, 318, 123, 393, 157, 488, 201, 598, 252, 732, 318, 887, 393, 1076, 488, 1291, 598, 1549, 732, 1845, 887, 2194, 1076, 2592, 1291, 3060, 1549, 3589, 1845, 4206, 2194, 4904, 2592, 5708, 3060, 6615, 3589

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(), 80); Coefficients(R!( 1/((1-x^9)*(&*[1-x^(2*j): j in [1..8]])) )); // G. C. Greubel, Feb 03 2020

Maple

seq(coeff(series(1/((1-x^9)*mul(1-x^(2*j), j=1..8)), x, n+1), x, n), n = 0..80); # G. C. Greubel, Feb 03 2020

Mathematica

CoefficientList[Series[1/((1-x^9)*Product[1-x^(2*j), {j,8}]), {x,0,80}], x] (* G. C. Greubel, Feb 03 2020 *)

PARI

Vec(1/((1-x^9)*prod(j=1,8,1-x^(2*j))) +O('x^80)) \\ G. C. Greubel, Feb 03 2020

Sage

def A266772_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 1/((1-x^9)*product(1-x^(2*j) for j in (1..8))) ).list()
A266772_list(80) # G. C. Greubel, Feb 03 2020

Formula

G.f.: 1/((1-t^2)*(1-t^4)*(1-t^6)*(1-t^8)*(1-t^9)*(1-t^10)*(1-t^12)*(1-t^14)*(1-t^16)).