A008645 Molien series of 6 X 6 upper triangular matrices over GF( 2 ).

1, 1, 2, 2, 4, 4, 6, 6, 10, 10, 14, 14, 20, 20, 26, 26, 36, 36, 46, 46, 60, 60, 74, 74, 94, 94, 114, 114, 140, 140, 166, 166, 202, 202, 238, 238, 284, 284, 330, 330, 390, 390, 450, 450, 524, 524, 598, 598

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Different from A018819 (see g.f.). - Joerg Arndt, Apr 22 2016

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 245
• 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, 1, -1, -1, 1, -1, 1, 1, -1, -1, 1, 1, -1, 1, -1, -1, 1, 1, -1, -1, 1, -1, 1, 1, -1, -1, 1, 1, -1, 1, -1, -1, 1, -1, 1, 1, -1, 1, -1, -1, 1, 1, -1, -1, 1, -1, 1, 1, -1).

Programs

Magma

R:=PowerSeriesRing(Integers(), 50); Coefficients(R!( 1/(&*[1-x^(2^j): j in [0..5]]) )); // G. C. Greubel, Feb 02 2020

Maple

seq(coeff(series( 1/mul((1-x^(2^j)), j=0..5)), x, n+1), x, n), n = 0..50); # G. C. Greubel, Feb 02 2020

Mathematica

CoefficientList[Series[1/(1-x)/(1-x^2)/(1-x^4)/(1-x^8)/(1-x^16)/(1-x^32), {x, 0, 100}], x] (* Vaclav Kotesovec, Apr 22 2016 *)

PARI

Vec( 1/prod(j=0,5, 1-x^(2^j)) +O('x^50) ) \\ G. C. Greubel, Feb 02 2020

Sage

def A008645_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 1/product(1-x^(2^j) for j in (0..5)) ).list()
A008645_list(50) # G. C. Greubel, Feb 02 2020

Formula

G.f.: 1/((1-x)*(1-x^2)*(1-x^4)*(1-x^8)*(1-x^16)*(1-x^32)).