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

0,5

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.

a(n) is the number of partitions of n into parts p, 2 <= p <= 9. - Joerg Arndt, Apr 04 2026

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

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)).

a(n) = floor((n^7 +154*n^6 +9667*n^5 +318010*n^4 +5906376*n^3 +62129648*n^2 +332226160*n)/1828915200 - (n mod 2)*(n^3+66*n^2+1299*n+3496)/18432 + ((n+2) mod 3)*(n^2+44*n)/2916 - ((2*n^2+n) mod 3)*n/729 + ((n^3+n^2+n+2) mod 4)*n/256 + 1). - Hoang Xuan Thanh, Apr 04 2026

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 *)

Prog

(PARI) Vec( prod(j=2, 9, 1/(1-x^j)) + O('x^70) ) \\ G. C. Greubel, Feb 01 2020
(Magma) R<x>:=PowerSeriesRing(Integers(), 70); Coefficients(R!( (&*[1/(1-x^j): j in [2..9]]) )); // G. C. Greubel, Feb 01 2020
(SageMath)
def A266777_list(prec):
    P.<x> = 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

Crossrefs

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

Sequence in context: A232047 A060029 A100471 * A248518 A095700 A339404

Adjacent sequences: A266774 A266775 A266776 * A266778 A266779 A266780

Keyword

nonn,easy

Author

N. J. A. Sloane, Jan 11 2016

Status

approved