A266779 Molien series for invariants of finite Coxeter group A_10.

1, 0, 1, 1, 2, 2, 4, 4, 7, 8, 12, 14, 20, 23, 32, 38, 50, 59, 77, 90, 115, 135, 168, 197, 243, 283, 344, 401, 481, 558, 665, 767, 906, 1043, 1221, 1401, 1631, 1862, 2155, 2454, 2823, 3203, 3668, 4147, 4727, 5330, 6047, 6798, 7685, 8612, 9700, 10843, 12168, 13566, 15178, 16877, 18825, 20884, 23226, 25707, 28517, 31489, 34842, 38396

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, -1, -1, -1, 0, 2, 3, 3, 3, 2, 1, 0, -2, -3, -4, -4, -5, -3, -1, 1, 3, 4, 5, 5, 4, 3, 1, -1, -3, -5, -4, -4, -3, -2, 0, 1, 2, 3, 3, 3, 2, 0, -1, -1, -1, -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..11]]) )); // G. C. Greubel, Feb 03 2020

Maple

seq(coeff(series(1/mul(1-x^j, j=2..11), x, n+1), x, n), n = 0..70); # G. C. Greubel, Feb 03 2020

Mathematica

CoefficientList[Series[1/Product[1-x^j, {j,2,11}], {x,0,70}], x] (* G. C. Greubel, Feb 03 2020 *)

PARI

Vec( 1/prod(j=2,11,1-x^j) +O('x^70)) \\ G. C. Greubel, Feb 03 2020

Sage

def A266779_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 1/product(1-x^j for j in (2..11))).list()
A266779_list(70) # G. C. Greubel, Feb 03 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)*(1-t^11)).