A266770 Molien series for invariants of finite Coxeter group D_7.

1, 0, 1, 0, 2, 0, 3, 1, 5, 1, 7, 2, 11, 3, 15, 5, 21, 7, 28, 11, 38, 15, 49, 21, 65, 28, 82, 38, 105, 49, 131, 65, 164, 82, 201, 105, 248, 131, 300, 164, 364, 201, 436, 248, 522, 300, 618, 364, 733, 436, 860, 522, 1009, 618, 1175, 733, 1367, 860, 1579, 1009, 1824, 1175, 2093, 1367, 2400, 1579, 2738, 1824, 3120, 2093, 3539, 2400, 4011

Offset: 0

Author

N. J. A. Sloane, Jan 10 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
• Index entries for linear recurrences with constant coefficients, signature (0,1,0,1,0,0,1,0,-1,-1,-1,0,0,-2,0,0,1,1,0,1,2,1,0,1,-1, 0,-1,-2,-1,0,-1,-1,0,0,2,0,0,1,1,1,0,-1,0,0,-1,0,-1,0,1).

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^7)*(&*[1-x^(2*j): j in [1..6]])) )); // G. C. Greubel, Jan 31 2020

Maple

seq(coeff(series(1/((1-x^7)*mul(1-x^(2*j), j=1..6)), x, n+1), x, n), n = 0..80); # G. C. Greubel, Jan 31 2020

Mathematica

CoefficientList[Series[1/((1-x^7)*Product[1-x^(2*j), {j,6}]), {x,0,80}], x] (* G. C. Greubel, Jan 31 2020 *)

PARI

Vec(1/((1-x^7)*prod(j=1,6,1-x^(2*j))) +O('x^80)) \\ G. C. Greubel, Jan 31 2020

Sage

def A266770_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 1/((1-x^7)*product(1-x^(2*j) for j in (1..6))) ).list()
A266770_list(80) # G. C. Greubel, Jan 31 2020

Formula

G.f.: 1/((1-x^2)*(1-x^4)*(1-x^6)*(1-x^7)*(1-x^8)*(1-x^10)*(1-x^12)).