A266775 Molien series for invariants of finite Coxeter group D_12 (bisected).

1, 1, 2, 3, 5, 7, 12, 16, 24, 33, 47, 63, 88, 115, 155, 202, 266, 341, 443, 560, 715, 897, 1129, 1401, 1746, 2146, 2645, 3228, 3941, 4771, 5781, 6948, 8353, 9979, 11913, 14144, 16785, 19814, 23374, 27454, 32211, 37645, 43954, 51130, 59417, 68827, 79631, 91863, 105857, 121645

Offset: 0

Author

N. J. A. Sloane, Jan 11 2016

Keywords

• nonn

Comments

The Molien series for the finite Coxeter group of type D_k (k >= 3) has g.f. = 1/Product_i (1-x^(1+m_i)) where the m_i are [1,3,5,...,2k-3,k-1]. If k is even only even powers of x appear, and we bisect the sequence.

References

• J. E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge, 1990. See Table 3.1, page 59.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for Molien series

Crossrefs

Molien series for finite Coxeter groups D_3 through D_12 are A266755, A266769, A266768, A003402, and A266770-A266775.

Programs

Magma

R:=PowerSeriesRing(Integers(), 50); Coefficients(R!( 1/((1-x^6)*(&*[1-x^j: j in [1..11]])) )); // G. C. Greubel, Jan 31 2020

Maple

S:=series(1/((1-x^6)*mul(1-x^j, j=1..11)), x, 55): seq(coeff(S, x, j), j=0..50); # G. C. Greubel, Jan 31 2020

Mathematica

CoefficientList[Series[1/((1-t^6)*Product[1-t^j, {j,11}]), {t,0,50}], t] (* G. C. Greubel, Jan 31 2020 *)

PARI

Vec( 1/( (1-x^6)*prod(j=1,11, 1-x^j) ) + O('x^50)) \\ G. C. Greubel, Jan 31 2020

Sage

[( 1/((1-x^6)*product(1-x^j for j in (1..11))) ).series(x, n+1).list()[n] for n in (0..50)] # G. C. Greubel, Jan 31 2020

Formula

G.f.: 1/((1-t^2)*(1-t^4)*(1-t^6)*(1-t^8)*(1-t^10)*(1-t^12)^2*(1-t^14)*(1-t^16)*(1-t^18)*(1-t^20)*(1-t^22)), bisected.

G.f.: 1/( (1-t^6)*Product_{j=1..11} (1-t^j) ). - G. C. Greubel, Feb 01 2020