%I #4 Apr 17 2014 14:29:45
%S 0,1,0,1,1,3,2,5,6,11,11,18,23,34,40,55,73,95,120,150,202,244,320,376,
%T 511,588,784,885,1205,1340,1802,1978,2691,2922,3938,4235,5745,6130,
%U 8255,8745,11815,12442,16709,17501,23531,24533,32820,34075,45581,47156
%N Number of partitions of n such that the number of odd parts is a part.
%e a(8) counts these 6 partitions: 521, 4211, 41111, 332, 3221, 22211.
%t z = 62; f[n_] := f[n] = IntegerPartitions[n];
%t Table[Count[f[n], p_ /; MemberQ[p, Count[Mod[p, 2], 0]]], {n, 0, z}] (* A240573 *)
%t Table[Count[f[n], p_ /; MemberQ[p, Count[Mod[p, 2], 1]]], {n, 0, z}] (* A240574 *)
%t Table[Count[f[n], p_ /; MemberQ[p, Count[Mod[p, 2], 0]] && MemberQ[p, Count[Mod[p, 2], 1]]], {n, 0, z}] (* A240575 *)
%t Table[Count[f[n], p_ /; MemberQ[p, Count[Mod[p, 2], 0]] || MemberQ[p, Count[Mod[p, 2], 1]]], {n, 0, z}] (* A240576 *)
%t Table[Count[f[n], p_ /; MemberQ[p, Count[Mod[p, 2], 0]] && ! MemberQ[p, Count[Mod[p, 2], 1]]], {n, 0, z}] (* A240577 *)
%t Table[Count[f[n], p_ /; ! MemberQ[p, Count[Mod[p, 2], 0]] && MemberQ[p, Count[Mod[p, 2], 1]]], {n, 0, z}] (* A240578 *)
%t Table[Count[f[n], p_ /; ! MemberQ[p, Count[Mod[p, 2], 0]] && ! MemberQ[p, Count[Mod[p, 2], 1]]], {n, 0, z}] (* A240579 *)
%Y Cf. A240573, A240575, A240576, A240577, A240578, A240579.
%K nonn,easy
%O 0,6
%A _Clark Kimberling_, Apr 10 2014