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A008651 Molien series of binary icosahedral group.

%I A008651 #43 Jul 08 2023 16:09:21
%S A008651 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,
%T A008651 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,
%U A008651 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
%N A008651 Molien series of binary icosahedral group.
%C A008651 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
%D A008651 T. A. Springer, Invariant Theory, Lecture Notes in Math., Vol. 585, Springer, p. 97.
%D A008651 S. Mukai, An Introduction to Invariants and Moduli, Cambridge, 2003; see p. 19.
%H A008651 G. C. Greubel, <a href="/A008651/b008651.txt">Table of n, a(n) for n = 0..1000</a>
%H A008651 Burnett Meyer, <a href="https://doi.org/10.4153/CJM-1954-016-2">On the symmetries of spherical harmonics</a>, Canadian Journal of Mathematics 6 (1954): 135-157.
%H A008651 <a href="/index/Mo#Molien">Index entries for Molien series</a>
%H A008651 <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (-1,0,1,1,1,1,0,-1,-1).
%F A008651 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
%F A008651 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
%F A008651 a(n) = -a(-1-n) for all n in Z. - _Michael Somos_, Dec 01 2014
%F A008651 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
%F A008651 G.f.: (1+x^15)/((1-x^10)*(1-x^6)) - not reduced _William Lionheart_, May 04 2019
%e A008651 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.
%e A008651 G.f. = 1 + x^6 + x^10 + x^12 + x^15 + x^16 + x^18 + x^20 + x^21 + x^22 + ...
%e A008651 G.f. = 1 + q^12 + q^20 + q^24 + q^30 + q^32 + q^36 + q^40 + q^42 + q^44 + ...
%p A008651 t1:=(1+x^10+x^20)/((1-x^6)*(1-x^15));
%p A008651 series(t1,x,100);
%p A008651 seriestolist(%);
%t A008651 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 *)
%t A008651 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 *)
%o A008651 (PARI) Vec(O(x^99)+(1+x^10+x^20)/((1-x^6)*(1-x^15))) \\ _M. F. Hasler_, Dec 01 2014
%o A008651 (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 */
%o A008651 (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 */
%o A008651 (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
%o A008651 (Sage)
%o A008651 def A008651_list(prec):
%o A008651     P.<x> = PowerSeriesRing(ZZ, prec)
%o A008651     return P((1+x^10+x^20)/((1-x^6)*(1-x^15))).list()
%o A008651 A008651_list(100) # _G. C. Greubel_, Sep 07 2019
%Y A008651 Cf. A319974 for harmonic polynomials in four variables invariant under a group.
%K A008651 nonn,easy
%O A008651 0,31
%A A008651 _N. J. A. Sloane_