A008652 Molien series for group of 3 X 3 upper triangular matrices over GF( 4 ).

1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 6, 6, 6, 6, 8, 8, 8, 8, 10, 10, 10, 10, 12, 12, 12, 12, 15, 15, 15, 15, 18, 18, 18, 18, 21, 21, 21, 21, 24, 24, 24, 24, 28, 28, 28, 28, 32, 32, 32, 32, 36, 36, 36, 36, 40, 40

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice

Comments

Partitions into parts 1, 4, and 16. - Joerg Arndt, Apr 29 2014

References

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

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 220
• Index entries for Molien series
• Index entries for linear recurrences with constant coefficients, signature (1, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1, 0, 0, -1, 1).

Programs

GAP

List([0..65], n-> Int((Int(n/4)+3)^2/8) ); # G. C. Greubel, Sep 07 2019~

Magma

[Floor((Floor(n/4)+3)^2/8): n in [0..65]]; // G. C. Greubel, Sep 07 2019

Maple

seq(coeff(series(1/((1-x)*(1-x^4)*(1-x^16)), x, n+1), x, n), n = 0..65); # G. C. Greubel, Sep 07 2019

Mathematica

Table[Floor[(Floor[n/4] + 3)^2/8], {n, 0, 61}] (* or *) Table[Floor[(n + 3)^2/8], {n, 0, 15}, {4}] // Flatten (* Jean-François Alcover, Jul 17 2013, updated Feb 26 2016 *)
LinearRecurrence[{1,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,-1,1}, {1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,6,6,6,6,8}, 70] (* Harvey P. Dale, Jan 30 2018 *)

PARI

a(n)=(n\4 + 3)^2\8 \\ Charles R Greathouse IV, Feb 10 2017

Sage

[floor((floor(n/4)+3)^2/8) for n in (0..65)] # G. C. Greubel, Sep 07 2019

Formula

G.f.: 1/((1-x)*(1-x^4)*(1-x^16)).

G.f.: 1/((1+x)^2*(1-x)^3*(1+x^2)^2*(1+x^4)*(1+x^8)). - Bruno Berselli, Jul 25 2013

a(n) ~ 1/128*n^2. - Ralf Stephan, Apr 29 2014