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Number of partitions of n such that the number of parts is a part and the number of distinct parts is not a part.
5

%I #4 Apr 27 2014 10:23:52

%S 0,0,0,0,1,1,0,2,1,3,3,6,6,10,12,18,21,28,35,48,56,78,93,115,143,187,

%T 219,282,337,419,496,629,736,912,1090,1324,1564,1901,2238,2720,3187,

%U 3821,4501,5387,6291,7455,8770,10341,12080,14227,16575,19479,22676

%N Number of partitions of n such that the number of parts is a part and the number of distinct parts is not a part.

%F a(n) + A241377(n) + A241378(n) = A241381(n) for n >= 0.

%e a(9) counts these 3 partitions: 51111, 4221, 333.

%t z = 30; f[n_] := f[n] = IntegerPartitions[n]; d[p_] := [p] = Length[DeleteDuplicates[p]];

%t Table[Count[f[n], p_ /; MemberQ[p, Length[p]] && MemberQ[p, d[p]]], {n, 0, z}] (* A241377 *)

%t Table[Count[f[n], p_ /; ! MemberQ[p, Length[p]] && MemberQ[p, d[p]]], {n, 0, z}] (* A241378 *)

%t Table[Count[f[n], p_ /; MemberQ[p, Length[p]] && ! MemberQ[p, d[p]]], {n, 0, z}] (* A241379 *)

%t Table[Count[f[n], p_ /; ! MemberQ[p, Length[p]] && ! MemberQ[p, d[p]]], {n, 0, z}] (* A241380 *)

%t Table[Count[f[n], p_ /; MemberQ[p, Length[p]] || MemberQ[p, d[p]]], {n, 0, z}] (* A241381 *)

%Y Cf. A241377, A241378, A241380, A241381.

%K nonn,easy

%O 0,8

%A _Clark Kimberling_, Apr 21 2014