%I
%S 0,0,0,0,1,1,2,2,4,4,6,7,10,11,15,17,22,25,31,37,45,51,62,73,87,98,
%T 116,133,159,182,211,241,276,322,369,419,479,539,622,705,807,909,1022,
%U 1163,1310,1483,1681,1880,2119,2365,2637,2947,3314,3756,4185,4644
%N Number of partitions p of n into distinct parts, including round(mean(p)).
%F a(n) + A241316(n) = A000009(n) for n >= 1.
%e a(10) counts these 6 partitions: 91, 82, 73, 721, 64, 631.
%t z = 30; f[n_] := f[n] = Select[IntegerPartitions[n], Max[Length /@ Split@#] == 1 &];
%t Table[Count[f[n], p_ /; MemberQ[p, Floor[Mean[p]]]], {n, 0, z}] (* A241312 *)
%t Table[Count[f[n], p_ /; ! MemberQ[p, Floor[Mean[p]]]], {n, 0, z}] (* A241313 *)
%t Table[Count[f[n], p_ /; MemberQ[p, Ceiling[Mean[p]]]], {n, 0, z}] (* A241314 *)
%t Table[Count[f[n], p_ /; ! MemberQ[p, Ceiling[Mean[p]]]], {n, 0, z}] (* A241315 *)
%t Table[Count[f[n], p_ /; MemberQ[p, Round[Mean[p]]]], {n, 0, z}] (* A241316 *)
%t Table[Count[f[n], p_ /; ! MemberQ[p, Round[Mean[p]]]], {n, 0, ] (* A241317 *)
%Y Cf. A241316, A241318, A000009.
%K nonn,easy
%O 0,7
%A _Clark Kimberling_, Apr 19 2014
