A008650 Molien series of 4 X 4 upper triangular matrices over GF( 3 ).

1, 1, 1, 2, 2, 2, 3, 3, 3, 5, 5, 5, 7, 7, 7, 9, 9, 9, 12, 12, 12, 15, 15, 15, 18, 18, 18, 23, 23, 23, 28, 28, 28, 33, 33, 33, 40, 40, 40, 47, 47, 47, 54, 54, 54, 63, 63, 63, 72, 72, 72, 81, 81, 81, 93, 93, 93, 105, 105

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

References

• D. J. Benson, Polynomial Invariants of Finite Groups, Cambridge, 1993, p. 105.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 235
• Index entries for Molien series
• Index entries for linear recurrences with constant coefficients, signature (1, 0, 1, -1, 0, 0, 0, 0, 1, -1, 0, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1, 0, -1, 1, 0, 0, 0, 0, -1, 1, 0, 1, -1).

Programs

Magma

R:=PowerSeriesRing(Integers(), 70); Coefficients(R!( 1/((1-x)*(1-x^3)*(1-x^9)*(1-x^27)) )); // G. C. Greubel, Sep 06 2019

Maple

1/((1-x)*(1-x^3)*(1-x^9)*(1-x^27)): seq(coeff(series(%, x, n+1), x, n), n=0..70);

Mathematica

CoefficientList[Series[1/((1-x)*(1-x^3)*(1-x^9)*(1-x^27)), {x,0,70}], x] (* G. C. Greubel, Sep 06 2019 *)

PARI

my(x='x+O('x^70)); Vec(1/((1-x)*(1-x^3)*(1-x^9)*(1-x^27))) \\ G. C. Greubel, Sep 06 2019

Sage

def A008650_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P(1/((1-x)*(1-x^3)*(1-x^9)*(1-x^27))).list()
A008650_list(70) # G. C. Greubel, Sep 06 2019

Formula

a(n) ~ 1/4374*n^3. - Ralf Stephan, Apr 29 2014

G.f.: 1/((1-x)*(1-x^3)*(1-x^9)*(1-x^27)).