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A007690
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Number of partitions of n in which no part occurs just once.
(Formerly M0167)
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24
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1, 0, 1, 1, 2, 1, 4, 2, 6, 5, 9, 7, 16, 11, 22, 20, 33, 28, 51, 42, 71, 66, 100, 92, 147, 131, 199, 193, 275, 263, 385, 364, 516, 511, 694, 686, 946, 925, 1246, 1260, 1650, 1663, 2194, 2202, 2857, 2928, 3721, 3813, 4866, 4967, 6257, 6487, 8051, 8342, 10369
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,5
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COMMENTS
| Euler transform of period 6 sequence [0,1,1,1,0,1,...]. - Michael Somos Apr 21 2004
Also number of partitions of n into parts, each larger than 1, such that consecutive integers do not both appear as parts. Example: a(6)=4 because we have [6],[4,2],[3,3] and [2,2,2]. - Emeric Deutsch, Feb 16 2006
Also number of partitions of n into parts divisible by 2 or 3. - Alexander E. Holroyd (holroyd(AT)math.ubc.ca), May 28 2008
Infinite convolution product of [1,0,1,1,1,1,1] aerated n-1 times. i.e. [1,0,1,1,1,1,1] * [1,0,0,0,1,0,1] * [1,0,0,0,0,0,1] * ... [From Mats Granvik, Aug 07 2009]
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REFERENCES
| G. E. Andrews, Number Theory, Dover Publications, 1994. page 197. MR1298627
R. Honsberger, Mathematical Gems III, M.A.A., 1985, p. 242.
George E. Andrews, The Theory of Partitions, Addison-Wesley, Reading, Mass., 1976, p. 14, Example 9.
G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976 (p. 14, Example 9).
P. A. MacMahon, Combinatory Analysis, Cambridge Univ. Press, London and New York, Vol. 1, 1915 and Vol. 2, 1916; see vol. 2, p 54, Article 300.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
| Alois P. Heinz, Table of n, a(n) for n = 0..1000
A. E. Holroyd, T. M. Liggett and D. Romik, Integrals, partitions and cellular automata
Eric Weisstein's World of Mathematics, Partition Function P
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FORMULA
| G.f.: Prod{k>0 is a multiple of 2 or 3} (1/(1-x^k)). - Christian G. Bower, Jun 23 2000
G.f.: product{i=1, oo, (1+x^3j)/(1-x^2j)} - Jon Perry, Mar 29 2004
G.f. is a period 1 Fourier series which satisfies f(-1 / (864 t)) = 1/6 (t/i)^(-1/2) g(t) where q = exp(2 pi i t) and g(t) is g.f. for A137566. - Michael Somos, Jan 26 2008
Contribution from Alois P. Heinz, Oct 09 2011: (Start)
a(n) = A000041(n) - A183558(n).
a(n) = A183568(n,0) - A183568(n,1).
G.f.: Product_{j>0} (1-x^j+x^(2*j))/(1-x^j). (End)
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EXAMPLE
| a(6) = 4 because we have [3,3],[2,2,2],[2,2,1,1] and [1,1,1,1,1,1].
q + q^49 + q^73 + 2*q^97 + q^121 + 4*q^145 + 2*q^169 + 6*q^193 + ...
1 + x^2 + x^3 + 2*x^4 + x^5 + 4*x^6 + 2*x^7 + 6*x^8 + 5*x^9 + 9*x^10 + ...
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MAPLE
| G:=product((1-x^j+x^(2*j))/(1-x^j), j=1..70): Gser:=series(G, x=0, 60): 1, seq(coeff(Gser, x^n), n=1..54); # Emeric Deutsch, Feb 10 2006
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PROG
| (PARI) a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff(eta(x^6+A)/eta(x^2+A)/eta(x^3+A), n)) /* Michael Somos Apr 21 2004 */
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CROSSREFS
| Cf. A000041, A055922, A055923, A114917, A114918, A183558, A183568.
Sequence in context: A004795 A161268 A176837 * A205685 A143375 A074364
Adjacent sequences: A007687 A007688 A007689 * A007691 A007692 A007693
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KEYWORD
| nonn
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), Robert G. Wilson v (rgwv(AT)rgwv.com)
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