A241413 - OEIS

%I #4 Apr 29 2014 00:06:39

%S 0,1,0,1,1,4,5,8,10,17,21,29,38,59,68,100,124,170,214,288,351,470,576,

%T 743,921,1176,1430,1816,2214,2753,3364,4176,5015,6215,7478,9120,10966,

%U 13351,15916,19301,22982,27618,32846,39354,46515,55570,65598,77842,91730

%N Number of partitions p of n such that the number of numbers having multiplicity 1 in p is a part of p.

%e a(6) counts these 5 partitions:  42, 411, 321, 3111, 21111; e.g., 411 is counted because 1 part of 411 has multiplicity 1, and 1 is a part of 411.

%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. A241414, A241415, A241416, A241417, A239737.

%K nonn,easy

%O 0,6

%A _Clark Kimberling_, Apr 23 2014