A266768 Molien series for invariants of finite Coxeter group D_5.

1, 0, 1, 0, 2, 1, 3, 1, 5, 2, 7, 3, 10, 5, 13, 7, 18, 10, 23, 13, 30, 18, 37, 23, 47, 30, 57, 37, 70, 47, 84, 57, 101, 70, 119, 84, 141, 101, 164, 119, 192, 141, 221, 164, 255, 192, 291, 221, 333, 255, 377, 291, 427, 333, 480, 377, 540, 427, 603, 480, 674, 540, 748, 603, 831, 674, 918, 748, 1014, 831, 1115, 918, 1226, 1014, 1342, 1115

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 (0,1,0,1,1,0,-1,0,-1,-2,0,0,0,0,2,1,0,1,0,-1,-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^2)*(1-x^4)*(1-x^6)*(1-x^5)*(1-x^8)) )); // G. C. Greubel, Jan 31 2020

Maple

seq(coeff(series(1/((1-x^2)*(1-x^4)*(1-x^6)*(1-x^5)*(1-x^8)), x, n+1), x, n), n = 0..80); # G. C. Greubel, Jan 31 2020

Mathematica

CoefficientList[Series[1/((1-x^2)*(1-x^4)*(1-x^6)*(1-x^5)*(1-x^8)), {x,0,80}], x] (* G. C. Greubel, Jan 31 2020 *)

PARI

my(x='x+O('x^80)); Vec(1/((1-x^2)*(1-x^4)*(1-x^6)*(1-x^5)*(1-x^8))) \\ G. C. Greubel, Jan 31 2020

Sage

def A266768_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 1/((1-x^2)*(1-x^4)*(1-x^6)*(1-x^5)*(1-x^8)) ).list()
A266768_list(80) # G. C. Greubel, Jan 31 2020

Formula

G.f.: 1/((1-x^2)*(1-x^4)*(1-x^6)*(1-x^5)*(1-x^8)).

a(n) = a(n-2)+a(n-4)+a(n-5)-a(n-7)-a(n-9)-2*a(n-10)+2*a(n-15)+a(n-16)+a(n-18)-a(n-20)-a(n-21)-a(n-23)+a(n-25). - Wesley Ivan Hurt, May 03 2021