A266780 Molien series for invariants of finite Coxeter group A_11.

1, 0, 1, 1, 2, 2, 4, 4, 7, 8, 12, 14, 21, 23, 33, 39, 52, 61, 81, 94, 122, 143, 180, 211, 264, 306, 377, 440, 533, 619, 746, 861, 1028, 1186, 1401, 1612, 1895, 2168, 2532, 2894, 3356, 3822, 4414, 5008, 5755, 6516, 7448, 8410, 9580, 10780, 12232, 13737, 15524, 17388, 19592, 21885, 24580, 27400, 30674, 34117, 38097, 42269, 47074, 52133

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, 0, -1, -1, 1, 2, 3, 3, 3, 2, 1, -1, -2, -3, -3, -5, -5, -4, -2, 0, 2, 4, 5, 6, 6, 5, 3, 2, -2, -3, -5, -6, -6, -5, -4, -2, 0, 2, 4, 5, 5, 3, 3, 2, 1, -1, -2, -3, -3, -3, -2, -1, 1, 1, 0, 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..12]]) )); // G. C. Greubel, Feb 04 2020

Maple

S:=series(1/mul(1-x^j, j=2..12)), x, 75):
seq(coeff(S, x, j), j=0..70); # G. C. Greubel, Feb 04 2020

Mathematica

CoefficientList[Series[1/Times@@(1-t^Range[2,12]),{t,0,70}],t] (* Harvey P. Dale, Jun 20 2017 *)

PARI

Vec( 1/prod(j=2,12, 1-x^j) +O('x^70) ) \\ G. C. Greubel, Feb 04 2020

Sage

def A266780_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 1/prod(1-x^j for j in (2..12)) ).list()
A266780_list(70) # G. C. Greubel, Feb 04 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)*(1-t^12)).