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

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice

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).

Crossrefs

Cf. A005795.

Programs

Magma

MolienSeries(CoxeterGroup("E7")); // Sergei Haller (sergei(AT)sergei-haller.de), Dec 21 2006

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 *)

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. = 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

Formula

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