A005795 -id:A005795 - cp's OEIS frontend

A005796 Degrees of fundamental invariants of Weyl group W(E_8).

Original entry on oeis.org

2, 8, 12, 14, 18, 20, 24, 30

Offset: 1

N. J. A. Sloane

a(n) - 1 is the n-th number < 30 that is relatively prime to 30. [Jonathan Sondow, Apr 16 2009]

H. S. M. Coxeter and W. O. J. Moser, Generators and Relations for Discrete Groups, 4th ed., Springer-Verlag, NY, reprinted 1984, p. 141.
G. J. Fox, Letter to N. J. A. Sloane, May 1991 [The notes on this letter reference this A-number but actually it contains a different sequence, which is not currently in the OEIS. - _Andrey Zabolotskiy_, Dec 09 2024]

Cf. A273867, A005795.

a(n) = Prime(n+2) + 1, for n = 2, 3, ..., 8. [Jonathan Sondow, Apr 16 2009]

A008583 Molien series for Weyl group E_7.

Original entry on oeis.org

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

N. J. A. Sloane

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.

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).

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).

Cf. A005795.

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

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

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 *)

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

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

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

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A005796 Degrees of fundamental invariants of Weyl group W(E_8).

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A008583 Molien series for Weyl group E_7.

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A273867 Degrees of fundamental invariants of Weyl group W(E_6).

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