A241444 - OEIS

%I #4 Apr 28 2014 17:53:30

%S 0,0,0,0,0,1,2,4,5,9,12,17,21,34,38,57,72,97,121,169,204,279,338,440,

%T 551,703,857,1095,1341,1664,2038,2536,3061,3777,4578,5564,6726,8170,

%U 9776,11849,14131,16992,20246,24264,28703,34322,40535,48156,56761,67239

%N Number of partitions of n such that the number of parts having multiplicity 1 is a part and the number of distinct parts is not a part.

%F a(n) + A241442(n) + A241443(n) = A241446(n) for n >= 0.

%e a(6) counts these 2 partitions:  411, 3111.

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

%t Table[Count[f[n], p_ /; MemberQ[p, u[p]] && MemberQ[p, d[p]]], {n, 0, z}]  (* A241442 *)

%t Table[Count[f[n], p_ /; ! MemberQ[p, u[p]] && MemberQ[p, d[p]] ], {n, 0,  z}] (* A241443 *)

%t Table[Count[f[n], p_ /; MemberQ[p, u[p]] && ! MemberQ[p, d[p]] ], {n, 0, z}] (* A241444 *)

%t Table[Count[f[n], p_ /; ! MemberQ[p, u[p]] && ! MemberQ[p, d[p]] ], {n, 0, z}] (* A241445 *)

%t Table[Count[f[n], p_ /; MemberQ[p, u[p]] || MemberQ[p, d[p]] ], {n, 0, z}] (* A241446 *)

%Y Cf. A241442, A241443, A241445, A241446.

%K nonn,easy

%O 0,7

%A _Clark Kimberling_, Apr 23 2014