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%I #9 Jul 07 2023 23:08:56
%S 1,1,0,0,2,0,0,0,0,7,0,0,0,0,0,0,34,0,0,0,0,0,0,0,0,192,0,0,0,0,0,0,0,
%T 0,0,0,1206,0,0,0,0,0,0,0,0,0,0,0,0,8033,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
%U 55974,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0
%N Number of integer partitions of n satisfying (length) = (mean). Partitions of n into sqrt(n) parts.
%F a(n^2) = A206240(n).
%e The a(0) = 1 through a(9) = 7 partitions:
%e () (1) . . (22) . . . . (333)
%e (31) (432)
%e (441)
%e (522)
%e (531)
%e (621)
%e (711)
%t Table[Length[If[n==0,{{}},Select[IntegerPartitions[n],Mean[#]==Length[#]&]]],{n,0,30}]
%Y The strict case is A107379(sqrt(n)).
%Y Without zeros we have A206240.
%Y These partitions have ranks A363951.
%Y A008284 counts partitions by length, A058398 by mean.
%Y A067538 counts partitions with integer mean, ranks A316413.
%Y Cf. A025065, A026905, A237984, A327472, A327482, A363944, A363949.
%K nonn
%O 0,5
%A _Gus Wiseman_, Jul 07 2023