A325488 Dimensions of space of harmonic polynomials of each degree invariant under the full icosahedral group.

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

Offset: 0

Author

William Lionheart, May 04 2019

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_h, the full icosahedral group including inversions, the generating function is 1/((1 - t^10)*(1 - t^6)).

Links

• Burnett Meyer, On the symmetries of spherical harmonics, Canadian Journal of Mathematics 6 (1954): 135-157.
• Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,-1).

Crossrefs

Cf. A008651 for the icosahedral rotation group which is derived from this sequence using Theorem 8 of Meyer, h(t,I)=(1+t^15)*h(t,I_h) as I_h has 15 symmetry planes.

Programs

Mathematica

CoefficientList[Series[(1 - t^10)^(-1) (1 - t^6)^(-1) , {t, 0, 100}],
  t]

PARI

Vec(1 / ((1 - x)^2*(1 + x)^2*(1 - x + x^2)*(1 + x + x^2)*(1 - x + x^2 - x^3 + x^4)*(1 + x + x^2 + x^3 + x^4)) + O(x^100)) \\ Colin Barker, Jun 26 2019

Formula

G.f.: 1/((1 - t^10)*(1 - t^6)).

a(n) = a(n-6) + a(n-10) - a(n-16) for n>15. - Colin Barker, Jun 26 2019