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Number of partitions p of n that contain a proper partition of the number of parts of p.
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%I #18 Dec 10 2016 22:38:41

%S 0,0,0,0,1,2,4,8,12,19,28,42,58,82,110,150,200,263,344,450,578,741,

%T 947,1197,1513,1899,2374,2954,3669,4529,5576,6849,8380,10223,12449

%N Number of partitions p of n that contain a proper partition of the number of parts of p.

%e a(9) counts these 12 partitions: [6,2,1], [5,2,1,1], [4,3,1,1], [4,2,2,1], [4,2,1,1,1], [3,3,1,1,1], [3,2,2,2], [3,2,2,1,1], [3,2,1,1,1,1], [2,2,2,2,1], [2,2,2,1,1,1],[2,2,1,1,1,1,1]; e.g., [2,2,1] is a proper partition of 5, which is the number of parts in [3,2,2,1,1].

%t Table[parts = IntegerPartitions[z];

%t Count[Table[rest = Rest[parts[[nn]]];

%t seq = Map[{#, Flatten[{___, #, ___}]} &,

%t Rest[IntegerPartitions[Length[rest] + 1]]];

%t Apply[Or, Map[MatchQ[rest, #[[2]]] &, seq]], {nn, Length[parts]}],

%t True], {z, 30}] (* _Peter J. C. Moses_, Dec 06 2016 *)

%Y Cf. A000041.

%K nonn,easy

%O 1,6

%A _Clark Kimberling_, Dec 07 2016