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A361656 Number of odd-length integer partitions of n with integer mean.

%I A361656 #11 Mar 24 2023 17:46:39
%S A361656 0,1,1,2,1,2,4,2,1,9,8,2,13,2,16,51,1,2,58,2,85,144,57,2,49,194,102,
%T A361656 381,437,2,629,2,1,956,298,2043,1954,2,491,2293,1116,2,4479,2,6752,
%U A361656 14671,1256,2,193,8035,4570,11614,22143,2,28585,39810,16476,24691,4566
%N A361656 Number of odd-length integer partitions of n with integer mean.
%C A361656 These are partitions of n whose length is an odd divisor of n.
%H A361656 Andrew Howroyd, <a href="/A361656/b361656.txt">Table of n, a(n) for n = 0..1000</a>
%e A361656 The a(1) = 1 through a(10) = 8 partitions (A = 10):
%e A361656   1   2   3     4   5       6     7         8   9           A
%e A361656           111       11111   222   1111111       333         22222
%e A361656                             321                 432         32221
%e A361656                             411                 441         33211
%e A361656                                                 522         42211
%e A361656                                                 531         43111
%e A361656                                                 621         52111
%e A361656                                                 711         61111
%e A361656                                                 111111111
%e A361656 For example, the partition (3,3,2,1,1) has length 5 and mean 2, so is counted under a(10).
%t A361656 Table[Length[Select[IntegerPartitions[n], OddQ[Length[#]]&&IntegerQ[Mean[#]]&]],{n,0,30}]
%o A361656 (PARI) a(n)=if(n==0, 0, sumdiv(n, d, if(d%2, polcoef(1/prod(k=1, d, 1 - x^k + O(x^(n-d+1))), n-d)))) \\ _Andrew Howroyd_, Mar 24 2023
%Y A361656 Odd-length partitions are counted by A027193, bisection A236559.
%Y A361656 Including even-length gives A067538 bisected, strict A102627, ranks A316413.
%Y A361656 The even-length version is counted by A361655.
%Y A361656 A000041 counts integer partitions, strict A000009.
%Y A361656 A027187 counts even-length partitions, bisection A236913.
%Y A361656 A051293 counts subsets with integer mean, median A000975.
%Y A361656 A058398 counts partitions by mean, see also A008284, A327482.
%Y A361656 A325347 counts partitions with integer median, complement A307683.
%Y A361656 A326567/A326568 gives mean of prime indices.
%Y A361656 A326622 counts factorizations with integer mean.
%Y A361656 Cf. A000016, A067659, A082550, A237984, A240219, A327475, A348551, A361653.
%K A361656 nonn
%O A361656 0,4
%A A361656 _Gus Wiseman_, Mar 24 2023