A239737 - OEIS

%I #19 Apr 24 2014 10:27:39

%S 0,1,1,2,2,5,7,10,13,21,28,38,54,77,99,137,180,236,306,398,504,644,

%T 807,1018,1278,1599,1972,2458,3039,3743,4592,5659,6884,8436,10235,

%U 12445,15021,18204,21842,26334,31501,37746,44956,53707,63657,75738,89536,106057

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

%F a(n) + A241417(n) = A000041(n) for n >= 0.

%e a(6) counts these 7 partitions:  42, 411, 321, 3111, 2211, 21111, 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, A241415, A241416, A241417, A000041.

%K nonn,easy

%O 0,4

%A _Clark Kimberling_, Apr 23 2014