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%I #4 Apr 29 2014 00:07:02
%S 0,0,1,1,1,1,2,2,3,4,7,9,16,18,31,37,56,66,92,110,153,174,231,275,357,
%T 423,542,642,825,990,1228,1483,1869,2221,2757,3325,4055,4853,5926,
%U 7033,8519,10128,12110,14353,17142,20168,23938,28215,33243,39019,45968
%N Number of partitions p of n such that the number of numbers having multiplicity 1 in p is not a part and the number of numbers having multiplicity > 1 is a part.
%F a(n) + A241415(n) + A241416(n) = A239737(n) for n >= 0.
%e a(6) counts these 2 partitions: 2211, 111111.
%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, A241416, A241417, A239737, A000041.
%K nonn,easy
%O 0,7
%A _Clark Kimberling_, Apr 23 2014
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