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A327780
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Number of integer partitions of n whose LCM is 2 * n.
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5
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0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 2, 12, 0, 0, 6, 0, 10, 32, 6, 0, 8, 0, 9, 0, 32, 0, 505, 0, 0, 108, 16, 147, 258, 0, 20, 170, 134, 0, 2030, 0, 140, 1865, 30, 0, 80, 0, 105, 350, 236, 0, 419, 500, 617, 474, 49, 0, 40966, 0, 56, 8225, 0, 785
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OFFSET
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0,15
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LINKS
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FORMULA
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a(n) = Sum_{d|2*n} mu(d)*([x^n] B(2*n/d, x)) for n > 0, where B(m,x) = 1/(Product_{d|m} 1 - x^d). - Andrew Howroyd, Feb 12 2022
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EXAMPLE
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The a(10) = 1 through a(20) = 10 partitions (A = 10) (empty columns not shown):
(541) (831) (7421) (A32) (9432) (A82)
(74111) (5532) (9441) (8552)
(6522) (94221) (A811)
(6531) (94311) (85421)
(A311) (942111) (85511)
(53322) (9411111) (852221)
(65211) (854111)
(532221) (8522111)
(533211) (85211111)
(651111) (851111111)
(5322111)
(53211111)
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MATHEMATICA
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Table[Length[Select[IntegerPartitions[n], LCM@@#==2*n&]], {n, 30}]
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PROG
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(PARI)
b(m, n)={my(d=divisors(m)); polcoef(1/prod(i=1, #d, 1 - x^d[i] + O(x*x^n)), n)}
a(n)={if(n<1, 0, sumdiv(2*n, d, moebius(d)*b(2*n/d, n)))} \\ Andrew Howroyd, Oct 09 2019
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CROSSREFS
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The Heinz numbers of these partitions are given by A327775.
Partitions whose LCM is a multiple of their sum are A327778.
Partitions whose LCM is equal to their sum are A074761.
Partitions whose LCM is greater than their sum are A327779.
Partitions whose LCM is less than their sum are A327781.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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