%I
%S 0,1,2,3,4,6,8,11,15,21,27,38,48,65,86,111,140,189,233,306,383,481,
%T 608,782,936,1186,1481,1828,2223,2793,3331,4144,5012,6079,7437,9051,
%U 10586,12970,15738,18851,22161,26885,31644,38188,45142,52983,63328,75823,87404
%N Number of partitions p of n including floor(mean(p)) as a part.
%F a(n) + A241335(n) = A000041(n) for n >= 0.
%e a(6) counts these 8 partitions: 6, 33, 321, 3111, 222, 2211, 21111, 111111.
%t z = 30; f[n_] := f[n] = IntegerPartitions[n];
%t Table[Count[f[n], p_ /; MemberQ[p, Floor[Mean[p]]]], {n, 0, z}] (* A241334 *)
%t Table[Count[f[n], p_ /; ! MemberQ[p, Floor[Mean[p]]]], {n, 0, z}] (* A241335 *)
%t Table[Count[f[n], p_ /; MemberQ[p, Ceiling[Mean[p]]]], {n, 0, z}] (* A241336 *)
%t Table[Count[f[n], p_ /; ! MemberQ[p, Ceiling[Mean[p]]]], {n, 0, z}] (* A241337 *)
%t Table[Count[f[n], p_ /; MemberQ[p, Round[Mean[p]]]], {n, 0, z}] (* A241338 *)
%t Table[Count[f[n], p_ /; ! MemberQ[p, Round[Mean[p]]]], {n, 0, z}] (* A241339 *)
%Y Cf. A241335, A241336, A241338, A000041, A241312.
%K nonn,easy
%O 0,3
%A _Clark Kimberling_, Apr 20 2014
