A008635 Molien series for alternating group Alt_12 (or A_12).

1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 100, 133, 172, 224, 285, 366, 460, 582, 725, 905, 1116, 1380, 1686, 2063, 2503, 3036, 3655, 4401, 5262, 6290, 7476, 8877, 10489, 12384, 14552, 17084, 19978, 23334, 27156, 31570, 36578, 42333, 48849

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

References

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

Links

• Sean A. Irvine, 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, 0, -1, 1, -2, -1, 0, 0, 1, 0, 1, 1, 2, 1, 0, 0, -1, -2, -3, -3, -1, -1, 1, 0, 3, 4, 3, 3, 1, 2, -2, -3, -3, -4, -3, -3, -2, 2, 1, 3, 3, 4, 3, 0, 1, -1, -1, -3, -3, -2, -1, 0, 0, 1, 2, 1, 1, 0, 1, 0, 0, -1, -2, 1, -1, 0, 0, 1, 1, -1).

Programs

Magma

R:=PowerSeriesRing(Integers(), 50); Coefficients(R!( (1+x^66)/(&*[1-x^j: j in [1..12]]) )); // G. C. Greubel, Feb 02 2020

Maple

seq(coeff(series( (1+x^66)/mul((1-x^j), j=1..12)), x, n+1), x, n), n = 0..50); # G. C. Greubel, Feb 02 2020

Mathematica

CoefficientList[Series[(1+x^66)/Product[(1-x^j), {j,12}], {x,0,50}], x] (* G. C. Greubel, Feb 02 2020 *)

PARI

Vec( (1+x^66)/prod(j=1,12, 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^66)/product(1-x^j for j in (1..12)) ).list()
A008631_list(70) # G. C. Greubel, Feb 02 2020

Formula

G.f.: (1+x^66)/((1-x)*(1-x^2)*(1-x^3)*(1-x^4)*(1-x^5)*(1-x^6)*(1-x^7)*(1-x^8)*(1-x^9)*(1-x^10)*(1-x^11)*(1-x^12)).

Extensions

More terms from Sean A. Irvine, Apr 01 2018