%I #11 Jul 26 2024 06:40:32

%S 0,0,0,1,0,1,2,2,3,3,5,5,7,7,10,12,14,16,19,23,27,33,37,45,51,60,68,

%T 82,94,108,123,143,165,188,214,246,282,318,362,412,469,527,597,675,

%U 764,858,965,1086,1223,1367,1530,1717,1923,2144,2393,2674,2981,3315

%N Number of partitions of n into distinct parts of which the number of even parts is a part.

%F a(n) + A240869(n) = A000009(n) for n >= 0.

%e a(10) counts these 5 partitions: 82, 721, 631, 541, 4321.

%t z = 70; f[n_] := f[n] = Select[IntegerPartitions[n], Max[Length /@ Split@#] == 1 &];

%t t1 = Table[Count[f[n], p_ /; MemberQ[p, Count[Mod[p, 2], 0]]], {n, 0, z}] (* A240862 *)

%t t2 = Table[Count[f[n], p_ /; MemberQ[p, Count[Mod[p, 2], 1]]], {n, 0, z}] (* A240863, *)

%t t3 = Table[Count[f[n], p_ /; MemberQ[p, Count[Mod[p, 2], 0]] && MemberQ[p, Count[Mod[p, 2], 1]]], {n, 0, z}] (* A240864 *)

%t t4 = Table[Count[f[n], p_ /; MemberQ[p, Count[Mod[p, 2], 0]] || MemberQ[p, Count[Mod[p, 2], 1]]], {n, 0, z}] (* A240865 *)

%t t5 = Table[Count[f[n], p_ /; MemberQ[p, Count[Mod[p, 2], 0]] && ! MemberQ[p, Count[Mod[p, 2], 1]]], {n, 0, z}] (* A240866 *)

%t t6 = Table[Count[f[n], p_ /; ! MemberQ[p, Count[Mod[p, 2], 0]] && MemberQ[p, Count[Mod[p, 2], 1]]], {n, 0, z}] (* A240867 *)

%t t7 = Table[Count[f[n], p_ /; ! MemberQ[p, Count[Mod[p, 2], 0]] && ! MemberQ[p, Count[Mod[p, 2], 1]]], {n, 0, z}] (* A240868 *)

%Y Cf. A240863, A240864, A240865, A240866, A240867, A240868; for analogous sequences for unrestricted partitions, see A240573-A240579.

%K nonn,easy

%O 0,7

%A _Clark Kimberling_, Apr 14 2014