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Number of partitions p of n such that the number of numbers p having multiplicity 1 in p is not a part and the number of numbers having multiplicity > 1 is not a part.
6

%I #4 Apr 29 2014 00:07:59

%S 1,0,1,1,3,2,4,5,9,9,14,18,23,24,36,39,51,61,79,92,123,148,195,237,

%T 297,359,464,552,679,822,1012,1183,1465,1707,2075,2438,2956,3433,4173,

%U 4851,5837,6837,8218,9554,11518,13396,16022,18697,22300,25923,30873,35838

%N Number of partitions p of n such that the number of numbers p having multiplicity 1 in p is not a part and the number of numbers having multiplicity > 1 is not a part.

%F a(n) + A239737(n) = A000041(n) for n >= 0.

%e a(6) counts these 4 partitions: 6, 51, 33, 222.

%t z = 30; f[n_] := f[n] = IntegerPartitions[n]; u[p_] := Length[DeleteDuplicates[Select[p, Count[p, #] == &]]]; e[q_] := Length[DeleteDuplicates[Select[q, Count[q, #] > 1 &]]]

%t Table[Count[f[n], p_ /; MemberQ[p, u[p]]], {n, 0, z}] (* A241413 *)

%t Table[Count[f[n], p_ /; MemberQ[p, u[p]] && MemberQ[p, e[p]]], {n, 0, z}] (* A241414 *)

%t Table[Count[f[n], p_ /; ! MemberQ[p, u[p]] && MemberQ[p, e[p]] ], {n, 0, z}] (* A241415 *)

%t Table[Count[f[n], p_ /; MemberQ[p, u[p]] && ! MemberQ[p, e[p]] ], {n, 0, z}] (* A241416 *)

%t Table[Count[f[n], p_ /; ! MemberQ[p, u[p]] && ! MemberQ[p, e[p]] ], {n, 0, z}] (* A241417 *)

%t Table[Count[f[n], p_ /; MemberQ[p, u[p]] || MemberQ[p, e[p]] ], {n, 0, z}] (* A239737 *)

%Y Cf. A241413, A241414, A241415, A241416, A239737, A000041.

%K nonn,easy

%O 0,5

%A _Clark Kimberling_, Apr 23 2014