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

0,31

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

Table of n, a(n) for n=0..100.

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).

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

Mathematica

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

Prog

(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

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.

Sequence in context: A340928 A355742 A267860 * A339813 A178111 A178112

Adjacent sequences: A325485 A325486 A325487 * A325489 A325490 A325491

Keyword

nonn,easy

Author

William Lionheart, May 04 2019

Status

approved