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

0,3

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

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

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 *)

Prog

(PARI) Vec( 1/( (1-x^6)*prod(j=1, 11, 1-x^j) ) + O('x^50)) \\ G. C. Greubel, Jan 31 2020
(Magma) R<x>:=PowerSeriesRing(Integers(), 50); Coefficients(R!( 1/((1-x^6)*(&*[1-x^j: j in [1..11]])) )); // 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

Crossrefs

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

Sequence in context: A078912 A105930 A122622 * A024790 A373297 A308271

Adjacent sequences: A266772 A266773 A266774 * A266776 A266777 A266778

Keyword

nonn

Author

N. J. A. Sloane, Jan 11 2016

Status

approved