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A015737
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Number of 3's in partitions of n into distinct parts.
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2
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0, 0, 1, 1, 1, 1, 1, 2, 3, 4, 4, 5, 6, 8, 10, 12, 14, 17, 20, 24, 29, 34, 40, 47, 55, 64, 75, 87, 101, 117, 135, 155, 179, 205, 235, 269, 307, 350, 399, 453, 514, 583, 660, 746, 843, 950, 1070, 1205, 1354, 1520, 1705, 1910, 2138, 2392
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OFFSET
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1,8
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LINKS
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Vincenzo Librandi and Vaclav Kotesovec, Table of n, a(n) for n = 1..10000 (first 70 terms from Vincenzo Librandi)
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FORMULA
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G.f.: x^3 * product(j >= 1, 1 + x^j)/(1 + x^3). - Emeric Deutsch, Apr 17 2006
Corresponding g.f. for "number of k's" is x^k/(1 + x^k) * prod(n >= 1, 1 + x^n). - Joerg Arndt, Feb 20 2014
a(n) ~ exp(Pi*sqrt(n/3)) / (8*3^(1/4)*n^(3/4)). - Vaclav Kotesovec, Oct 30 2015
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EXAMPLE
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a(9) = 3 because in the eight partitions of 9 into distinct parts, namely [9], [8, 1], [7, 2], [6, 3], [6, 2, 1], [5, 4], [5, 3, 1] and [4, 3, 2], only three of those contain 3.
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MAPLE
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g:=x^3*product(1+x^j, j=1..60)/(1+x^3): gser:=series(g, x=0, 57): seq(coeff(gser, x, n), n=1..54); # Emeric Deutsch, Apr 17 2006
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MATHEMATICA
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Table[Count[Select[IntegerPartitions[n], Length[Union[#]] == Length[#] &], _?(MemberQ[#, 3] &)], {n, 60}] (* Harvey P. Dale, Aug 19 2011 *)
nmax = 100; Rest[CoefficientList[Series[x^3/(1 + x^3) * Product[1 + x^k, {k, 1, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Oct 30 2015 *)
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CROSSREFS
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Cf. A000009.
Sequence in context: A065328 A049877 A029063 * A015745 A017854 A261171
Adjacent sequences: A015734 A015735 A015736 * A015738 A015739 A015740
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KEYWORD
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nonn
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AUTHOR
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Clark Kimberling
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EXTENSIONS
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Clarified example, Harvey P. Dale, Aug 19 2011
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STATUS
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approved
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