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A008672
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Expansion of 1/((1-x)(1-x^3)(1-x^5)).
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2
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1, 1, 1, 2, 2, 3, 4, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 17, 18, 20, 22, 23, 25, 27, 29, 31, 33, 35, 37, 40, 42, 44, 47, 49, 52, 55, 57, 60, 63, 66, 69, 72, 75, 78, 82, 85, 88, 92, 95, 99, 103, 106, 110, 114, 118, 122, 126, 130, 134, 139, 143, 147, 152, 156, 161, 166
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,4
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COMMENTS
| Number of partitions of n into odd parts less than or equal 5.
1/((1-x^2)(1-x^6)(1-x^10)) is the Molien series for the icosahedral group [3,5] of order 120.
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REFERENCES
| L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 120, D(n;1,3,5).
W. Ebeling, Lattices and Codes, Vieweg; 2nd ed., 2002, see p. 164 etc.
F. Hirzebruch, Letter to N. J. A. Sloane, quoted in Ges. Abh. II, 796-798.
F. Klein, Lectures on the Icosahedron ..., 2nd Rev. Ed., 1913; reprinted by Dover, NY, 1956; see pp. 236-243.
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.
L. Smith, Polynomial Invariants of Finite Groups, Peters, 1995, p. 199 (No. 23).
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LINKS
| Vincenzo Librandi, Table of n, a(n) for n = 0..10000
G. Nebe, E. M. Rains and N. J. A. Sloane, Self-Dual Codes and Invariant Theory, Springer, Berlin, 2006.
G. E. Andrews, P. Paule, A. Riese and V. Strehl, MacMahon's partition analysis V. Bijections, recursions and magic squares
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 218
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 to sequences with linear recurrences with constant coefficients, signature (1,0,1,-1,1,-1,0,-1,1).
Index entries for Molien series
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FORMULA
| a(n) = round((n+3)*(n+6)/30).
a(n) = floor[n^2/30+3n/10+1].
G.f.: 1/((1-x)(1-x^3)(1-x^5)).
a(n)=a(-9-n). - Michael Somos Nov 16 2005
a(0)=1, a(1)=1, a(2)=1, a(3)=2, a(4)=2, a(5)=3, a(6)=4, a(7)=4, a(8)=5, a(n)=a(n-1)+a(n-3)-a(n-4)+a(n-5)-a(n-6)-a(n-8)+a(n-9) [From Harvey P. Dale, Feb 07 2012]
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MATHEMATICA
| CoefficientList[Series[1/((1-x)(1-x^3)(1-x^5)), {x, 0, 70}], x] (* or *) LinearRecurrence[{1, 0, 1, -1, 1, -1, 0, -1, 1}, {1, 1, 1, 2, 2, 3, 4, 4, 5}, 70] (* From Harvey P. Dale, Feb 07 2012 *)
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PROG
| (PARI) a(n)=(n^2+9*n)\30+1
(MAGMA) [Round((n+3)*(n+6)/30): n in [0..60]]; // Vincenzo Librandi, Jun 23 2011
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CROSSREFS
| a(n) = A025799(2n).
Sequence in context: A050294 A097950 A011885 * A097923 A027582 A011880
Adjacent sequences: A008669 A008670 A008671 * A008673 A008674 A008675
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KEYWORD
| nonn,nice,easy,changed
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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