A266781 Molien series for invariants of finite Coxeter group A_12.

1, 0, 1, 1, 2, 2, 4, 4, 7, 8, 12, 14, 21, 24, 33, 40, 53, 63, 83, 98, 126, 150, 188, 223, 278, 327, 401, 473, 573, 672, 809, 944, 1126, 1312, 1551, 1800, 2118, 2446, 2859, 3295, 3829, 4395, 5086, 5817, 6699, 7642, 8760, 9961, 11380, 12898, 14678, 16596, 18819, 21217, 23987, 26971, 30397, 34099, 38316, 42877, 48058, 53649, 59972, 66811

Offset: 0

Author

N. J. A. Sloane, Jan 11 2016

Keywords

• nonn
• easy

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, 0, -1, 0, 1, 2, 3, 3, 3, 2, 0, -1, -2, -3, -4, -4, -5, -4, -3, -1, 1, 3, 5, 7, 7, 6, 5, 3, 2, -1, -4, -6, -7, -8, -7, -6, -4, -1, 2, 3, 5, 6, 7, 7, 5, 3, 1, -1, -3, -4, -5, -4, -4, -3, -2, -1, 0, 2, 3, 3, 3, 2, 1, 0, -1, 0, 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..13]]) )); // G. C. Greubel, Feb 04 2020

Maple

S:=series(1/mul(1-x^j, j=2..13)), x, 75):
seq(coeff(S, x, j), j=0..70); # G. C. Greubel, Feb 04 2020

Mathematica

CoefficientList[Series[1/Product[1-x^j, {j,2,13}], {x,0,70}], x] (* G. C. Greubel, Feb 04 2020 *)
LinearRecurrence[{0,1,1,1,0,0,-1,-1,-1,-1,-1,0,0,-1,0,1,2,3,3,3,2,0,-1,-2,-3,-4,-4,-5,-4,-3,-1,1,3,5,7,7,6,5,3,2,-1,-4,-6,-7,-8,-7,-6,-4,-1,2,3,5,6,7,7,5,3,1,-1,-3,-4,-5,-4,-4,-3,-2,-1,0,2,3,3,3,2,1,0,-1,0,0,-1,-1,-1,-1,-1,0,0,1,1,1,0,-1},{1,0,1,1,2,2,4,4,7,8,12,14,21,24,33,40,53,63,83,98,126,150,188,223,278,327,401,473,573,672,809,944,1126,1312,1551,1800,2118,2446,2859,3295,3829,4395,5086,5817,6699,7642,8760,9961,11380,12898,14678,16596,18819,21217,23987,26971,30397,34099,38316,42877,48058,53649,59972,66811,74499,82813,92136,102204,113455,125613,139140,153754,169979,187481,206857,227767,250835,275713,303108,332617,365036,399950,438201,479372,524403,572813,625657,682451,744307,810735},80] (* Harvey P. Dale, Jul 01 2021 *)

PARI

Vec( 1/prod(j=2,13, 1-x^j) +O('x^70) ) \\ G. C. Greubel, Feb 04 2020

Sage

def A266781_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 1/prod(1-x^j for j in (2..13)) ).list()
A266781_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)*(1-t^13)).