A008643 Molien series for group of 4 X 4 upper triangular matrices over GF(2).

1, 1, 2, 2, 4, 4, 6, 6, 10, 10, 14, 14, 20, 20, 26, 26, 35, 35, 44, 44, 56, 56, 68, 68, 84, 84, 100, 100, 120, 120, 140, 140, 165, 165, 190, 190, 220, 220, 250, 250, 286, 286, 322, 322, 364, 364, 406, 406, 455, 455, 504, 504, 560, 560, 616, 616, 680, 680, 744

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Number of partitions of n into parts 1, 2, 4 and 8. - Ilya Gutkovskiy, May 24 2017

References

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

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 233
• Index entries for Molien series
• Index entries for linear recurrences with constant coefficients, signature (1,1,-1,1,-1,-1,1,1,-1,-1,1,-1,1,1,-1).

Crossrefs

Cf. A008804, A088932 (partial sums).

Programs

Magma

R:=PowerSeriesRing(Integers(), 65); Coefficients(R!( (&*[1/(1-x^(2^j)): j in [0..3]]) )); // G. C. Greubel, Feb 01 2020

Maple

a:= proc(n) local m, r; m := iquo(n, 8, 'r'); r:= iquo(r,2)+1; ([11, 17, 26, 35][r]+ (9+ 3*r+ 4*m) *m) *m/3+ [1, 2, 4, 6][r] end: seq(a(n), n=0..100);  # Alois P. Heinz, Oct 06 2008

Mathematica

CoefficientList[1/((1-x)*(1-x^2)*(1-x^4)*(1-x^8)) + O[x]^65, x] (* Jean-François Alcover, May 29 2015 *)
LinearRecurrence[{1,1,-1,1,-1,-1,1,1,-1,-1,1,-1,1,1,-1}, {1,1,2,2,4,4,6,6,10,10,14,14,20,20,26}, 65] (* Ray Chandler, Jul 15 2015 *)

PARI

my(x='x+O('x^65)); Vec(1/((1-x)*(1-x^2)*(1-x^4)*(1-x^8))) \\ G. C. Greubel, Feb 01 2020

PARI

my(b(m) = (m^3 + 12*m^2 + (44 - 3*(m%2))*m + 48)\48); vector(59,n,b((n-1)\2)) \\ Hoang Xuan Thanh, Aug 14 2025

Sage

def A077952_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 1/((1-x)*(1-x^2)*(1-x^4)*(1-x^8)) ).list()
A077952_list(65) # G. C. Greubel, Feb 01 2020

Formula

G.f.: 1/((1-x)*(1-x^2)*(1-x^4)*(1-x^8)).

a(n) = floor(((n+14)*(3*(n+1)*(-1)^n + 2*n^2 + 17*n + 57) + 24*(floor(n/2) + 1)*(-1)^floor(n/2))/768). - Tani Akinari, Jun 16 2013

a(n) ~ 1/384*n^3. - Ralf Stephan, Apr 29 2014