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%I #11 Jan 21 2023 19:44:52
%S 0,1,1,2,1,2,3,2,1,5,5,2,5,2,8,18,1,2,19,2,24,41,20,2,9,44,31,94,102,
%T 2,125,2,1,206,68,365,382,2,98,433,155,2,716,2,1162,2332,196,2,17,
%U 1108,563,1665,3287,2,3906,5474,2005,3083,509,2,9029
%N Number of odd-length integer partitions of n whose parts have the same mean as median.
%C The length and median of such a partition are integers with product n.
%H Andrew Howroyd, <a href="/A359895/b359895.txt">Table of n, a(n) for n = 0..1000</a>
%F a(p) = 2 for prime p. - _Andrew Howroyd_, Jan 21 2023
%e The a(1) = 1 through a(9) = 5 partitions:
%e (1) (2) (3) (4) (5) (6) (7) (8) (9)
%e (111) (11111) (222) (1111111) (333)
%e (321) (432)
%e (531)
%e (111111111)
%e The a(15) = 18 partitions:
%e (15)
%e (5,5,5)
%e (6,5,4)
%e (7,5,3)
%e (8,5,2)
%e (9,5,1)
%e (3,3,3,3,3)
%e (4,3,3,3,2)
%e (4,4,3,2,2)
%e (4,4,3,3,1)
%e (5,3,3,2,2)
%e (5,3,3,3,1)
%e (5,4,3,2,1)
%e (5,5,3,1,1)
%e (6,3,3,2,1)
%e (6,4,3,1,1)
%e (7,3,3,1,1)
%e (1,1,1,1,1,1,1,1,1,1,1,1,1,1,1)
%t Table[Length[Select[IntegerPartitions[n], OddQ[Length[#]]&&Mean[#]==Median[#]&]],{n,0,30}]
%o (PARI) \\ P(n, k, m) is g.f. for k parts of max size m.
%o P(n, k, m)={polcoef(1/prod(i=1, m, 1 - y*x^i + O(x*x^n)), k, y)}
%o a(n)={if(n==0, 0, sumdiv(n, d, if(d%2, my(m=n/d, h=d\2, r=n-m*(h+1)+h); polcoef(P(r, h, m)*P(r, h, r), r))))} \\ _Andrew Howroyd_, Jan 21 2023
%Y This is the odd-length case of A240219, complement A359894, strict A359897.
%Y These partitions are ranked by A359891, complement A359892.
%Y The complement is counted by A359896.
%Y The strict case is A359899, complement A359900.
%Y The version for factorizations is A359910.
%Y A000041 counts partitions, strict A000009.
%Y A008284/A058398/A327482 count partitions by mean, ranked by A326567/A326568.
%Y A027193 counts odd-length partitions, strict A067659, ranked by A026424.
%Y A067538 counts ptns with integer mean, strict A102627, ranked by A316413.
%Y A237984 counts ptns containing their mean, strict A240850, ranked by A327473.
%Y A325347 counts ptns with integer median, strict A359907, ranked by A359908.
%Y A359893 and A359901 count partitions by median, odd-length A359902.
%Y Cf. A008289, A316313, A327472, A327475, A327482, A359889, A359906.
%K nonn
%O 0,4
%A _Gus Wiseman_, Jan 20 2023
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