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A008651
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Molien series of binary icosahedral group.
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5
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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
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OFFSET
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0,31
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COMMENTS
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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
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REFERENCES
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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.
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LINKS
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FORMULA
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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
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
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EXAMPLE
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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 + ...
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MAPLE
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t1:=(1+x^10+x^20)/((1-x^6)*(1-x^15));
series(t1, x, 100);
seriestolist(%);
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MATHEMATICA
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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 *)
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PROG
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(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 */
(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
(Sage)
P.<x> = PowerSeriesRing(ZZ, prec)
return P((1+x^10+x^20)/((1-x^6)*(1-x^15))).list()
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CROSSREFS
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Cf. A319974 for harmonic polynomials in four variables invariant under a group.
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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