A008631 Molien series for alternating group Alt_8 (or A_8).

1, 1, 2, 3, 5, 7, 11, 15, 22, 29, 40, 52, 70, 89, 116, 146, 186, 230, 288, 352, 434, 525, 638, 764, 919, 1090, 1297, 1527, 1802, 2105, 2464, 2860, 3324, 3835, 4428, 5081, 5834, 6659, 7604, 8640, 9819, 11107, 12566, 14158, 15951, 17904, 20093, 22474, 25133

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
• Index entries for Molien series
• Index entries for linear recurrences with constant coefficients, signature (1,1,0,1,-2,-1,-1,-1,1,1,2,3,0,-1,-1,-4,-1,-1,0,3,2,1,1,-1,-1,-1,-2,1,0,1,1,-1).

Crossrefs

Different from A008637.

Programs

Magma

R:=PowerSeriesRing(Integers(), 50); Coefficients(R!( (1+x^28)/(&*[1-x^j: j in [1..8]]) )); // G. C. Greubel, Feb 02 2020

Maple

seq(coeff(series( (1+x^28)/mul((1-x^j), j=1..8)), x, n+1), x, n), n = 0..50); # G. C. Greubel, Feb 02 2020

Mathematica

CoefficientList[Series[(1+x^28)/Product[(1-x^j), {j,1,8}], {x,0,50}], x] (* G. C. Greubel, Feb 02 2020 *)
LinearRecurrence[{1,1,0,1,-2,-1,-1,-1,1,1,2,3,0,-1,-1,-4,-1,-1,0,3,2,1,1,-1,-1,-1,-2,1,0,1,1,-1},{1,1,2,3,5,7,11,15,22,29,40,52,70,89,116,146,186,230,288,352,434,525,638,764,919,1090,1297,1527,1802,2105,2464,2860},70] (* Harvey P. Dale, May 12 2022 *)

PARI

Vec( (1+x^28)/prod(j=1,8, 1-x^j) +O('x^50) ) \\ G. C. Greubel, Feb 02 2020

Sage

def A008631_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1+x^28)/product(1-x^j for j in (1..8)) ).list()
A008631_list(70) # G. C. Greubel, Feb 02 2020

Formula

G.f.: (1+x^28)/((1-x)*(1-x^2)*(1-x^3)*(1-x^4)*(1-x^5)*(1-x^6)*(1-x^7)*(1-x^8)).