A008649 Molien series of 3 X 3 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, 22, 22, 22, 26, 26, 26, 30, 30, 30, 35, 35, 35, 40, 40, 40, 45, 45, 45, 51, 51, 51, 57, 57, 57, 63, 63, 63, 70, 70, 70, 77, 77, 77, 84, 84, 84, 92, 92, 92, 100, 100, 100

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Number of partitions of n into parts 1, 3 or 9. - Reinhard Zumkeller, Aug 12 2011

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 219
• 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).

Programs

Magma

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

Maple

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

Mathematica

CoefficientList[Series[1/((1-x)*(1-x^3)*(1-x^9)), {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))) \\ G. C. Greubel, Sep 06 2019

Sage

def A008649_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P(1/((1-x)*(1-x^3)*(1-x^9))).list()
A008649_list(70) # G. C. Greubel, Sep 06 2019

Formula

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

a(n) = floor((6*(floor(n/3) +1)*(3*floor(n/3) -n +1) +n^2 +13*n +58)/54). - Tani Akinari, Jul 12 2013