A008613 Molien series for 3-dimensional representation of A_5.

1, 0, 1, 0, 1, 0, 2, 0, 2, 0, 3, 0, 4, 0, 4, 1, 5, 1, 6, 1, 7, 2, 8, 2, 9, 3, 10, 4, 11, 4, 13, 5, 14, 6, 15, 7, 17, 8, 18, 9, 20, 10, 22, 11, 23, 13, 25, 14, 27, 15, 29, 17, 31, 18, 33, 20, 35, 22, 37, 23, 40, 25, 42, 27

Offset

0,7

Comments

Also arises in connection with Lee weight enumerators of codes over GF(5).

Partitions of n into (any number of) parts 2, 6, and 10, and at most one part 15. - Joerg Arndt, May 15 2011

The Neusel and Smith reference on Example 4 (T. Molien) on the rotation group of an icosahedron is a representation of A_5. - Michael Somos, Feb 01 2018

References

D. J. Benson, Polynomial Invariants of Finite Groups, Cambridge, 1993, p. 101.

H. Derksen and G. Kemper, Computational Invariant Theory, Springer, 2002; p. 92.

G. van der Geer, Hilbert Modular Surfaces, Springer-Verlag, 1988; p. 192.

F. Klein, Lectures on the Icosahedron ..., 2nd Rev. Ed., 1913; reprinted by Dover, NY, 1956; see pp. 236-243.

F. Klein, Werke, II, p. 354.

M. D. Neusel and L. Smith, Invariant Theory of Finite Groups, AMS, 2010, p. 55.

Links

T. D. Noe, Table of n, a(n) for n = 0..1000

Roberto De Maria Nunes Mendes, Symmetries of spherical harmonics, Transactions of the American Mathematical Society 204 (1975): 161-178. See subgroup 109.

J. S. Leon, V. S. Pless and N. J. A. Sloane, Self-dual codes over GF(5), J. Combin. Theory, A 32 (1982), 178-194.

G. Nebe, E. M. Rains and N. J. A. Sloane, Self-Dual Codes and Invariant Theory, Springer, Berlin, 2006.

F. J. MacWilliams, C. L. Mallows and N. J. A. Sloane, Generalizations of Gleason's theorem on weight enumerators of self-dual codes, IEEE Trans. Inform. Theory, 18 (1972), 794-805; see p. 802, col. 2, foot.

Index entries for Molien series

Index entries for linear recurrences with constant coefficients, signature (-1,1,2,1,0,0,-1,-2,-1,1,1)

Formula

G.f.: (1+x^15)/((1-x^2)*(1-x^6)*(1-x^10)) = ( -1-x+x^3+x^5+x^4-x^8-x^7 ) / ( (1+x+x^2) *(x^4+x^3+x^2+x+1) *(1+x)^2 *(x-1)^3 ).

a(n) = -a(n-1)+a(n-2)+2*a(n-3)+a(n-4)-a(n-7)-2*a(n-8)-a(n-9)+a(n-10)+a (n-11), n>10. - Harvey P. Dale, May 15 2011

a(n) ~ 1/120*n^2. - Ralf Stephan, Apr 29 2014

a(n) = floor((n^2+3*n+105)/120+(n+1)*(-1)^n/8). - Tani Akinari, Sep 30 2014

Euler transform of length 30 sequence [0, 1, 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, Sep 30 2014

a(n) = a(-3-n) for all n in Z.

0 = a(n) - a(n+2) - a(n+6) + a(n+8) - [mod(n, 5) == 2] for all n in Z. - Michael Somos, Sep 30 2014

Example

G.f. = 1 + x^2 + x^4 + 2*x^6 + 2*x^8 + 3*x^10 + 4*x^12 + 4*x^14 + x^15 + ...

Maple

(1+x^15)/((1-x^2)*(1-x^6)*(1-x^10));

Mathematica

CoefficientList[Series[(1+x^15)/((1-x^2)(1-x^6)(1-x^10)), {x, 0, 100}], x] (* or *) LinearRecurrence[{-1, 1, 2, 1, 0, 0, -1, -2, -1, 1, 1}, {1, 0, 1, 0, 1, 0, 2, 0, 2, 0, 3}, 100] (* Harvey P. Dale, May 15 2011 *)
a[ n_] := Module[{m = If[ n < 0, -3 - n, n]}, m = If[ OddQ[m], m - 15, m]/2; SeriesCoefficient[ 1 / ((1 - x^1) (1 - x^3) (1 - x^5)), {x, 0, m}]]; (* Michael Somos, Feb 01 2018 *)
LinearRecurrence[{-1, 1, 2, 1, 0, 0, -1, -2, -1, 1, 1}, {1, 0, 1, 0, 1, 0, 2, 0, 2, 0, 3}, 80] (* Harvey P. Dale, Jul 09 2019 *)

Prog

(PARI) a(n)=(n^2 + 3*n + 105 + 15*(n+1)*(-1)^n)\120 \\ Charles R Greathouse IV, Feb 10 2017

Crossrefs

Sequence in context: A025805 A029192 A128619 * A165685 A035457 A005868

Adjacent sequences: A008610 A008611 A008612 * A008614 A008615 A008616

Keyword

nonn,easy,nice

Author

N. J. A. Sloane

Status

approved