|
%I #14 Feb 12 2022 20:28:57
%S 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,
%T 0,0,108,16,147,258,0,20,170,134,0,2030,0,140,1865,30,0,80,0,105,350,
%U 236,0,419,500,617,474,49,0,40966,0,56,8225,0,785
%N Number of integer partitions of n whose LCM is 2 * n.
%H Andrew Howroyd, <a href="/A327780/b327780.txt">Table of n, a(n) for n = 0..1000</a>
%F 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
%e The a(10) = 1 through a(20) = 10 partitions (A = 10) (empty columns not shown):
%e (541) (831) (7421) (A32) (9432) (A82)
%e (74111) (5532) (9441) (8552)
%e (6522) (94221) (A811)
%e (6531) (94311) (85421)
%e (A311) (942111) (85511)
%e (53322) (9411111) (852221)
%e (65211) (854111)
%e (532221) (8522111)
%e (533211) (85211111)
%e (651111) (851111111)
%e (5322111)
%e (53211111)
%t Table[Length[Select[IntegerPartitions[n],LCM@@#==2*n&]],{n,30}]
%o (PARI)
%o b(m,n)={my(d=divisors(m)); polcoef(1/prod(i=1, #d, 1 - x^d[i] + O(x*x^n)), n)}
%o a(n)={if(n<1, 0, sumdiv(2*n, d, moebius(d)*b(2*n/d, n)))} \\ _Andrew Howroyd_, Oct 09 2019
%Y The Heinz numbers of these partitions are given by A327775.
%Y Partitions whose LCM is a multiple of their sum are A327778.
%Y Partitions whose LCM is equal to their sum are A074761.
%Y Partitions whose LCM is greater than their sum are A327779.
%Y Partitions whose LCM is less than their sum are A327781.
%Y Cf. A018818, A290103, A316413, A326842.
%K nonn
%O 0,15
%A _Gus Wiseman_, Sep 25 2019
|
