A266778 Molien series for invariants of finite Coxeter group A_9.

1, 0, 1, 1, 2, 2, 4, 4, 7, 8, 12, 13, 20, 22, 31, 36, 48, 55, 73, 83, 107, 123, 154, 177, 220, 251, 306, 351, 422, 481, 575, 652, 771, 875, 1024, 1158, 1348, 1518, 1754, 1973, 2265, 2538, 2901, 3241, 3684, 4109, 4646, 5167, 5823, 6457, 7246, 8020, 8965, 9898, 11031, 12150, 13495, 14837, 16428, 18022, 19905, 21789, 23999, 26228, 28813

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, -1, -2, -1, 0, 1, 3, 3, 3, 2, 1, 0, -1, -4, -4, -4, -3, -2, 0, 2, 3, 4, 4, 4, 1, 0, -1, -2, -3, -3, -3, -1, 0, 1, 2, 1, 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..10]] )); // G. C. Greubel, Feb 02 2020

Maple

seq(coeff(series( mul(1/(1-x^j), j=2..10), x, n+1), x, n), n = 0..70); # G. C. Greubel, Feb 02 2020

Mathematica

CoefficientList[Series[Product[1/(1-x^j), {j,2,10}], {x,0,70}], x] (* G. C. Greubel, Feb 02 2020 *)
LinearRecurrence[{0,1,1,1,0,0,-1,-1,-1,-1,-2,-1,0,1,3,3,3,2,1,0,-1,-4,-4,-4,-3,-2,0,2,3,4,4,4,1,0,-1,-2,-3,-3,-3,-1,0,1,2,1,1,1,1,0,0,-1,-1,-1,0,1},{1,0,1,1,2,2,4,4,7,8,12,13,20,22,31,36,48,55,73,83,107,123,154,177,220,251,306,351,422,481,575,652,771,875,1024,1158,1348,1518,1754,1973,2265,2538,2901,3241,3684,4109,4646,5167,5823,6457,7246,8020,8965,9898},70] (* Harvey P. Dale, Aug 10 2021 *)

PARI

Vec( prod(j=2,10, 1/(1-x^j)) +O('x^70) ) \\ G. C. Greubel, Feb 02 2020

Sage

def A266778_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( product(1/(1-x^j) for j in (2..10)) ).list()
A266778_list(70) # G. C. Greubel, Feb 02 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)*(1-t^10)).