A266776 Molien series for invariants of finite Coxeter group A_7.

1, 0, 1, 1, 2, 2, 4, 4, 7, 7, 11, 12, 18, 19, 27, 30, 40, 44, 58, 64, 82, 91, 113, 126, 155, 171, 207, 230, 274, 303, 358, 395, 462, 509, 589, 649, 746, 818, 934, 1024, 1161, 1269, 1432, 1562, 1753, 1909, 2131, 2317, 2577, 2794, 3095, 3352, 3698, 3997, 4396, 4743, 5200, 5601, 6121, 6584, 7177, 7705, 8377, 8983, 9741, 10429, 11285, 12065

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 into parts 2, 3, ..., 8. - Joerg Arndt, Apr 05 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,-2,-2,-1,1,2,2,3,2,1,-1,-2,-3,-2,-2,-1,1,2,2,1,1,0,0,-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)).

a(n) = floor((n^6 +105*n^5 +4340*n^4 +91000*n^3 +1037232*n^2 +5906320*n)/29030400 - (n mod 2)*(2*n^3+105*n^2+1653*n+2710)/18432 + ((n+2) mod 3)*n/162 + ((n^2+n+2) mod 4)*n/256 + 1). - Hoang Xuan Thanh, Apr 04 2026

Mathematica

CoefficientList[Series[1/Product[1-t^k, {k, 2, 8}], {t, 0, 40}], t] (* G. C. Greubel, Oct 24 2018 *)

Prog

(PARI) t='t+O('t^40); Vec(1/prod(k=2, 8, 1-t^k)) \\ G. C. Greubel, Oct 24 2018
(Magma) m:=40; R<t>:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/(&*[1-t^k: k in [2..8]]))); // G. C. Greubel, Oct 24 2018

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: A197122 A064410 A304178 * A371514 A363214 A062896

Adjacent sequences: A266773 A266774 A266775 * A266777 A266778 A266779

Keyword

nonn,easy

Author

N. J. A. Sloane, Jan 11 2016

Status

approved