A008651 Molien series of binary icosahedral group.

1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 2, 2, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 3, 2, 2, 2, 2, 2, 3, 2, 2, 2, 3, 2, 3, 2, 2, 3, 3, 2, 3, 2, 3, 3, 3, 2, 3, 3, 3, 3, 3, 2, 4, 3, 3, 3, 3, 3, 4, 3, 3, 3, 4, 3, 4, 3, 3, 4

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Meyer's generating function h(t,G) generates the sequence of the dimensions of the spaces of G-invariant harmonic polynomials of each degree, where G is a point group on three-dimensional Euclidean space. For G=I, the icosahedral rotation group, the generating function gives rise to this sequence. See Table 1, p. 143. - William Lionheart, May 04 2019

Examples

			The Molien series is (1+q^20+q^40)/((1-q^12)*(1-q^30)). Since every other term would be zero, we replace q^2 with x to get the sequence.
G.f. = 1 + x^6 + x^10 + x^12 + x^15 + x^16 + x^18 + x^20 + x^21 + x^22 + ...
G.f. = 1 + q^12 + q^20 + q^24 + q^30 + q^32 + q^36 + q^40 + q^42 + q^44 + ...
		

References

• T. A. Springer, Invariant Theory, Lecture Notes in Math., Vol. 585, Springer, p. 97.
• S. Mukai, An Introduction to Invariants and Moduli, Cambridge, 2003; see p. 19.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Burnett Meyer, On the symmetries of spherical harmonics, Canadian Journal of Mathematics 6 (1954): 135-157.
• Index entries for Molien series
• Index entries for linear recurrences with constant coefficients, signature (-1,0,1,1,1,1,0,-1,-1).

Crossrefs

Cf. A319974 for harmonic polynomials in four variables invariant under a group.

Programs

Magma

I:=[1,0,0,0,0,0,1,0,0]; [n le 9 select I[n] else -Self(n-1) +Self(n-3)+Self(n-4)+Self(n-5)+Self(n-6)-Self(n-8)-Self(n-9): n in [1..100]]; // Vincenzo Librandi, Jun 24 2015

Maple

t1:=(1+x^10+x^20)/((1-x^6)*(1-x^15));
series(t1,x,100);
seriestolist(%);

Mathematica

a[ n_] := With[ {s = Boole[ n<0 ], m = If[ n<0, -1-n, n]}, (-1)^s * SeriesCoefficient[(1+x^15)/((1-x^6)*(1-x^10)), {x, 0 ,m}]]; (* Michael Somos, Dec 01 2014 *)
LinearRecurrence[{-1,0,1,1,1,1,0,-1,-1}, {1,0,0,0,0,0,1,0,0}, 100] (* Harvey P. Dale, May 04 2017 *)

PARI

Vec(O(x^99)+(1+x^10+x^20)/((1-x^6)*(1-x^15))) \\ M. F. Hasler, Dec 01 2014

PARI

{a(n) = my(s=n<0); if(s, n = -1-n); (-1)^s * polcoeff( (1 + x^15) / ( (1 - x^6) * (1 - x^10) ) + x * O(x^n), n)}; /* Michael Somos, Dec 01 2014 */

PARI

{a(n) = (n\30) + [0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1][n%30 + 1]}; /* Michael Somos, Dec 01 2014 */

Sage

def A008651_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((1+x^10+x^20)/((1-x^6)*(1-x^15))).list()
A008651_list(100) # G. C. Greubel, Sep 07 2019

Formula

G.f.: (1 +x -x^3 -x^4 -x^5 +x^7 +x^8)/((1+x)*(1-x)^2*(1+x+x^2)*(1+x+x^2+x^3+x^4)). - R. J. Mathar, Dec 01 2014

Euler transform of length 30 sequence [ 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1]. - Michael Somos, Dec 01 2014

a(n) = -a(-1-n) for all n in Z. - Michael Somos, Dec 01 2014

0 = 1 + a(n) + 2*a(n+1) + 2*a(n+2) + a(n+3) - a(n+5) - 2*a(n+6) - 2*a(n+7) - a(n+8) for all n in Z. - Michael Somos, Dec 01 2014

G.f.: (1+x^15)/((1-x^10)*(1-x^6)) - not reduced William Lionheart, May 04 2019