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A327780
Number of integer partitions of n whose LCM is 2 * n.
5
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
OFFSET
0,15
LINKS
FORMULA
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
EXAMPLE
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)
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], LCM@@#==2*n&]], {n, 30}]
PROG
(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
CROSSREFS
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.
Sequence in context: A110982 A079331 A265490 * A350258 A364781 A166489
KEYWORD
nonn
AUTHOR
Gus Wiseman, Sep 25 2019
STATUS
approved