

A008613


Molien series for 3dimensional representation of A_5.


1



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
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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, SpringerVerlag, 1988; p. 192.
F. Klein, Lectures on the Icosahedron ..., 2nd Rev. Ed., 1913; reprinted by Dover, NY, 1956; see pp. 236243.
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): 161178. See subgroup 109.
J. S. Leon, V. S. Pless and N. J. A. Sloane, Selfdual codes over GF(5), J. Combin. Theory, A 32 (1982), 178194.
G. Nebe, E. M. Rains and N. J. A. Sloane, SelfDual 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 selfdual codes, IEEE Trans. Inform. Theory, 18 (1972), 794805; 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)/((1x^2)*(1x^6)*(1x^10)) = ( 1x+x^3+x^5+x^4x^8x^7 ) / ( (1+x+x^2) *(x^4+x^3+x^2+x+1) *(1+x)^2 *(x1)^3 ).
a(n) = a(n1)+a(n2)+2*a(n3)+a(n4)a(n7)2*a(n8)a(n9)+a(n10)+a (n11), 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(3n) 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)/((1x^2)*(1x^6)*(1x^10));


MATHEMATICA

CoefficientList[Series[(1+x^15)/((1x^2)(1x^6)(1x^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



