A008583 Molien series for Weyl group E_7.

1, 1, 1, 2, 3, 4, 6, 8, 10, 14, 18, 22, 29, 36, 44, 55, 67, 80, 98, 117, 138, 165, 194, 226, 266, 309, 356, 413, 475, 542, 622, 708, 802, 911, 1029, 1157, 1304, 1462, 1633, 1827, 2036, 2261, 2514, 2785

Offset

0,4

Comments

The relevant generating function 1/((1-z^2)*(1-z^6)*(1-z^8)*(1-z^10)*(1-z^12)*(1-z^14)*(1-z^18)) is reduced with z^2=x below to indicate that the intermediate zeros are not stored in this sequence.

References

H. S. M. Coxeter and W. O. J. Moser, Generators and Relations for Discrete Groups, Ergebnisse der Mathematik und Ihrer Grenzgebiete, New Series, no. 14. Springer Verlag, 1957, Table 10.

L. Smith, Polynomial Invariants of Finite Groups, Peters, 1995, p. 199 (No. 36).

Links

T. D. Noe, Table of n, a(n) for n = 0..1000

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 250

Tengiz O. Gogoberidze, Baker's dozen digits of two sums involving reciprocal products of an integer and its greatest prime factor, arXiv:2407.12047 [math.GM], 2024. See p. 6.

Index entries for Molien series

Index entries for linear recurrences with constant coefficients, signature (1, 0, 1, 0, 0, 0, -1, -1, 0, -1, 0, 1, 0, 2, 0, 1, 0, 0, -1, 0, -2, 0, -1, 0, 1, 0, 1, 1, 0, 0, 0, -1, 0, -1, 1).

Formula

G.f.: 1/((1-x)*(1-x^3)*(1-x^4)*(1-x^5)*(1-x^6)*(1-x^7)*(1-x^9)).

Maple

A008583_list := proc(n) local G, j;
G:= series(1/((1-x)*(1-x^3)*(1-x^4)*(1-x^5)*(1-x^6)*(1-x^7)*(1-x^9)), x, n+1);
[seq(coeff(G, x, j), j=0..n)];
end proc; # Robert Israel, Mar 26 2012

Mathematica

CoefficientList[Series[1/((1-x)(1-x^3)(1-x^4)(1-x^5)(1-x^6)(1-x^7)(1-x^9)), {x, 0, 50}], x] (* Harvey P. Dale, Mar 04 2013 *)

Prog

(Magma) MolienSeries(CoxeterGroup("E7")); // Sergei Haller (sergei(AT)sergei-haller.de), Dec 21 2006
(PARI) A008583_list(n)=Vec(1/((1-x)*(1-x^3)*(1-x^4)*(1-x^5)*(1-x^6)*(1-x^7)*(1-x^9))+O(x^n))  /* returns n terms [a(0), ..., a(n-1)] */ \\ M. F. Hasler, Mar 26 2012
(Sage)
def A008583_list(n) :
    R.<t> = PowerSeriesRing(ZZ)
    G = 1/((1-t)*(1-t^3)*(1-t^4)*(1-t^5)*(1-t^6)*(1-t^7)*(1-t^9) + O(t^n))
    return G.padded_list()  # Peter Luschny, Mar 27 2012

Crossrefs

Cf. A005795.

Sequence in context: A003107 A217123 A014977 * A053253 A095913 A376622

Adjacent sequences: A008580 A008581 A008582 * A008584 A008585 A008586

Keyword

nonn,easy,nice

Author

N. J. A. Sloane

Status

approved