A266777 Molien series for invariants of finite Coxeter group A_8.

1, 0, 1, 1, 2, 2, 4, 4, 7, 8, 11, 13, 19, 21, 29, 34, 44, 51, 66, 75, 95, 110, 134, 155, 189, 215, 258, 296, 349, 398, 468, 529, 617, 698, 804, 907, 1042, 1167, 1332, 1492, 1690, 1886, 2130, 2366, 2660, 2951, 3298, 3649, 4069, 4484, 4981, 5482, 6064, 6657, 7347, 8041, 8849, 9670, 10605, 11565, 12659, 13769, 15034, 16330, 17782, 19278, 20955

Offset: 0

Author

N. J. A. Sloane, Jan 11 2016

Keywords

• nonn

Comments

The Molien series for the finite Coxeter group of type A_k (k >= 1) has g.f. = 1/Product_{i=2..k+1} (1-x^i).

Note that this is the root system A_k, not the alternating group Alt_k.

References

• J. E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge, 1990. See Table 3.1, page 59.

Links

• Ray Chandler, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (0,1,1,1,0,0,-1,-1,-1,-2,-2,0,1,2,3,3,2,1,0,-2,-3,-4,-3,-2,0,1,2,3,3,2,1,0,-2,-2,-1,-1,-1,0,0,1,1,1,0,-1).
• Index entries for Molien series

Crossrefs

Molien series for finite Coxeter groups A_1 through A_12 are A059841, A103221, A266755, A008667, A037145, A001996, and A266776-A266781.

Programs

Magma

R:=PowerSeriesRing(Integers(), 70); Coefficients(R!( (&*[1/(1-x^j): j in [2..9]]) )); // G. C. Greubel, Feb 01 2020

Maple

seq(coeff(series( mul(1/(1-x^j), j=2..9), x, n+1), x, n), n = 0..70); # G. C. Greubel, Feb 01 2020

Mathematica

CoefficientList[Series[Product[1/(1-x^j), {j,2,9}], {x,0,70}], x] (* G. C. Greubel, Feb 01 2020 *)

PARI

Vec( prod(j=2,9, 1/(1-x^j)) + O('x^70) ) \\ G. C. Greubel, Feb 01 2020

Sage

def A266777_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( product(1/(1-x^j) for j in (2..9)) ).list()
A266777_list(70) # G. C. Greubel, Feb 01 2020

Formula

G.f.: 1/((1-t^2)*(1-t^3)*(1-t^4)*(1-t^5)*(1-t^6)*(1-t^7)*(1-t^8)*(1-t^9)).