

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
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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 n1 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



FORMULA

G.f.: Product_{j>=1} (1x^(2j)+x^(3j))/(1x^j).
G.f. for the number of partitions of n having no part that appears exactly m times is Product_{k>0} (1/(1x^k)x^(m*k)).


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((1x^(2*j)+x^(3*j))/(1x^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/(1x^i)x^(2i), {i, 1, nn}], {x, 0, nn}], x] (* Geoffrey Critzer, Sep 30 2013 *)


CROSSREFS



KEYWORD

nonn


AUTHOR



STATUS

approved



