|
|
A266775
|
|
Molien series for invariants of finite Coxeter group D_12 (bisected).
|
|
10
|
|
|
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
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
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
|
|
|
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
|
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|