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A361655 Number of even-length integer partitions of 2n with integer mean.

%I A361655 #12 Mar 24 2023 17:58:56
%S A361655 0,1,3,4,10,6,33,8,65,68,117,12,583,14,319,1078,1416,18,3341,20,8035,
%T A361655 5799,1657,24,36708,16954,3496,24553,68528,30,192180,32,178802,91561,
%U A361655 14625,485598,955142,38,29223,316085,2622697,42,3528870,44,2443527,5740043
%N A361655 Number of even-length integer partitions of 2n with integer mean.
%H A361655 Andrew Howroyd, <a href="/A361655/b361655.txt">Table of n, a(n) for n = 0..1000</a>
%e A361655 The a(0) = 0 through a(5) = 6 partitions:
%e A361655   .  (11)  (22)    (33)      (44)        (55)
%e A361655            (31)    (42)      (53)        (64)
%e A361655            (1111)  (51)      (62)        (73)
%e A361655                    (111111)  (71)        (82)
%e A361655                              (2222)      (91)
%e A361655                              (3221)      (1111111111)
%e A361655                              (3311)
%e A361655                              (4211)
%e A361655                              (5111)
%e A361655                              (11111111)
%e A361655 For example, the partition (4,2,1,1) has length 4 and mean 2, so is counted under a(4).
%t A361655 Table[Length[Select[IntegerPartitions[2n], EvenQ[Length[#]]&&IntegerQ[Mean[#]]&]],{n,0,15}]
%o A361655 (PARI) a(n)=if(n==0, 0, sumdiv(n, d, polcoef(1/prod(k=1, 2*d, 1 - x^k + O(x*x^(2*(n-d)))), 2*(n-d)))) \\ _Andrew Howroyd_, Mar 24 2023
%Y A361655 Even-length partitions are counted by A027187, bisection A236913.
%Y A361655 Including odd-length partitions gives A067538 bisected, ranks A316413.
%Y A361655 For median instead of mean we have A361653.
%Y A361655 The odd-length version is counted by A361656.
%Y A361655 A000041 counts integer partitions, strict A000009.
%Y A361655 A051293 counts subsets with integer mean, median A000975.
%Y A361655 A058398 counts partitions by mean, see also A008284, A327482.
%Y A361655 A325347 counts partitions with integer median, complement A307683.
%Y A361655 A326567/A326568 gives mean of prime indices.
%Y A361655 A326622 counts factorizations with integer mean.
%Y A361655 Cf. A000016, A027193, A067659, A082550, A102627, A237984, A240219, A327475, A348551, A359893.
%K A361655 nonn
%O A361655 0,3
%A A361655 _Gus Wiseman_, Mar 23 2023
%E A361655 Terms a(36) and beyond from _Andrew Howroyd_, Mar 24 2023