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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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62
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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
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OFFSET
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0,5
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COMMENTS
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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] * ... . - Mats Granvik, Aug 07 2009
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REFERENCES
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G. E. Andrews, Number Theory, Dover Publications, 1994. page 197. MR1298627
G. E. Andrews, The Theory of Partitions, Addison-Wesley, Reading, Mass., 1976, p. 14, Example 9.
I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983, (2.5.6).
R. Honsberger, Mathematical Gems III, M.A.A., 1985, p. 242.
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
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FORMULA
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G.f.: Product_{k>0 is a multiple of 2 or 3} (1/(1-x^k)). - Christian G. Bower, Jun 23 2000
G.f.: Product_{j>=1} (1+x^(3*j)) / (1-x^(2*j)). - Jon Perry, Mar 29 2004
Euler transform of period 6 sequence [0, 1, 1, 1, 0, 1, ...]. - Michael Somos, Apr 21 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 the g.f. for A137566. - Michael Somos, Jan 26 2008
G.f.: Product_{j>0} (1-x^j+x^(2*j)) / (1-x^j). (End)
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EXAMPLE
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a(6) = 4 because we have [3,3], [2,2,2], [2,2,1,1] and [1,1,1,1,1,1].
G.f. = 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 + ...
G.f. = q + q^49 + q^73 + 2*q^97 + q^121 + 4*q^145 + 2*q^169 + 6*q^193 + ...
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MAPLE
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G:= mul((1-x^j+x^(2*j))/(1-x^j), j=1..70): Gser:=series(G, x, 60): seq(coeff(Gser, x, n), n=0..54); # Emeric Deutsch, Feb 10 2006
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MATHEMATICA
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nn=40; CoefficientList[Series[Product[1/(1-x^i)-x^i, {i, 1, nn}], {x, 0, nn}], x] (* Geoffrey Critzer, Dec 02 2012 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ x^6] / (QPochhammer[ x^2] QPochhammer[ x^3]), {x, 0, n}]; (* Michael Somos, Feb 22 2015 *)
nmax = 60; CoefficientList[Series[Product[(1 + x^(3*k))/(1 - x^(2*k)), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 23 2015 *)
Table[Length@Select[Tally /@ IntegerPartitions@n, AllTrue[#, Last[#] > 1 &] &], {n, 0, 54}] (* Robert Price, Aug 17 2020 *)
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PROG
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(PARI) {a(n) = my(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
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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