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 A116645 Number of partitions of n having no doubletons. By a doubleton in a partition we mean an occurrence of a part exactly twice (the partition [4,(3,3),2,2,2,(1,1)] of 18 has two doubletons, shown between parentheses). 12
 1, 1, 1, 3, 3, 5, 8, 10, 13, 20, 26, 33, 46, 58, 75, 101, 125, 157, 206, 253, 317, 403, 494, 608, 760, 926, 1131, 1393, 1685, 2038, 2487, 2985, 3585, 4331, 5168, 6172, 7392, 8771, 10410, 12382, 14622, 17258, 20400, 23975, 28159, 33115, 38739, 45298, 53000 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS Number of partitions of n having no part that appears exactly twice. Infinite convolution product of [1,1,0,1,1,1,1,1,1,1] aerated n-1 times. I.e., [1,1,0,1,1,1,1,1,1,1] * [1,0,1,0,0,0,1,0,1,0] * [1,0,0,1,0,0,0,0,0,1] * ... - Mats Granvik, Gary W. Adamson, Aug 07 2009 LINKS Alois P. Heinz, Table of n, a(n) for n = 0..1000 FORMULA G.f.: Product_{j>=1} (1-x^(2j)+x^(3j))/(1-x^j). G.f. for the number of partitions of n having no part that appears exactly m times is Product_{k>0} (1/(1-x^k)-x^(m*k)). a(n) = A000041(n) - A183559(n) = A183568(n,0) - A183568(n,2). - Alois P. Heinz, Oct 09 2011 EXAMPLE a(4) = 3 because we have [4],[3,1] and [1,1,1,1] (the partitions [2,2] and [2,1,1] do not qualify since each of them has a doubleton). MAPLE h:=product((1-x^(2*j)+x^(3*j))/(1-x^j), j=1..60): hser:=series(h, x=0, 60): seq(coeff(hser, x, n), n=0..56); MATHEMATICA nn=48; CoefficientList[Series[Product[1/(1-x^i)-x^(2i), {i, 1, nn}], {x, 0, nn}], x] (* Geoffrey Critzer, Sep 30 2013 *) CROSSREFS Column 0 of A116644. Cf. A000041, A007690, A116595, A183559, A183568. Sequence in context: A015723 A333150 A342343 * A177739 A323581 A327731 Adjacent sequences: A116642 A116643 A116644 * A116646 A116647 A116648 KEYWORD nonn AUTHOR Emeric Deutsch and Vladeta Jovovic, Feb 20 2006 STATUS approved

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Last modified September 25 05:35 EDT 2023. Contains 365582 sequences. (Running on oeis4.)