A241410 - OEIS

%I #7 Feb 11 2019 09:32:00

%S 0,1,0,1,0,1,2,2,4,5,7,8,12,17,22,29,33,49,59,77,97,123,153,199,234,

%T 306,375,460,557,708,845,1048,1266,1548,1852,2282,2698,3303,3919,4732,

%U 5634,6786,7991,9598,11343,13502,15897,18912,22180,26298,30775,36259

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

%C As used here, the term "distinct parts" includes each number, once, that occurs more than once; e.g., the distinct parts of the partition {4,3,3,1,1,1} are 4, 3, 1.

%e a(6) counts these 2 partitions: 42, 321.

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

%t Table[Count[f[n], p_ /; MemberQ[p, e[p]]], {n, 0, z}]  (* A241408 *)

%t Table[Count[f[n], p_ /; MemberQ[p, e[p]] && MemberQ[p, d[p]]], {n, 0, z}]  (* A241409 *)

%t Table[Count[f[n], p_ /; ! MemberQ[p, e[p]] && MemberQ[p, d[p]] ], {n, 0, z}] (* A241410 *)

%t Table[Count[f[n], p_ /; MemberQ[p, e[p]] && ! MemberQ[p, d[p]] ], {n, 0, z}] (* A241411  *)

%t Table[Count[f[n], p_ /; ! MemberQ[p, e[p]] && ! MemberQ[p, d[p]] ], {n, 0, z}] (* A241412  *)

%Y Cf. A241408, A241409, A241411, A241412.

%K nonn,easy

%O 0,7

%A _Clark Kimberling_, Apr 22 2014