%I #6 Dec 07 2016 10:20:47

%S 0,0,0,1,1,4,4,10,12,21,25,46,50,82,99,148,175,259,303,435,513,708,

%T 845,1146,1347,1802,2138,2793,3318,4273,5050,6471,7621,9641,11406,

%U 14210,16758,20833,24475,30143

%N Number of partitions p of n that contain a proper partition of the maximal part of p.

%e a(8) counts these 10 partitions: [4,3,1], [4,2,2], [4,2,1,1], [4,1,1,1,1], [3,2,2,1], [3,2,1,1,1], [3,1,1,1,1,1], [2,2,2,1,1], [2,2,1,1,1,1],[2,1,1,1,1,1,1]; e.g., [3,1] is a proper partition of 4.

%t Table[parts = IntegerPartitions[z]; parts = Drop[parts,

%t Position[Map[#[[1]] &, parts], Floor[z/2], 1, 1][[1]][[1]] - 1];

%t Count[Table[{first, rest} = {First[#], Rest[#]} &[parts[[nn]]];

%t Apply[Or, Map[MatchQ[rest, #] &, Map[Flatten[{___, #, ___}] &,

%t Rest[IntegerPartitions[first]]]]], {nn, Length[parts]}], True], {z, 30}]

%t (* _Peter J. C. Moses_, Dec 02 2016 *)

%Y Cf. A000041.

%K nonn,easy

%O 1,6

%A _Clark Kimberling_, Dec 04 2016