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%I #20 Oct 05 2019 11:34:06
%S 0,1,2,3,4,5,7,10,13,18,25,31,44,57,73,99,127,166,205,268,349,424,548,
%T 674,858,1075,1324,1665,1973,2472,3085,3774,4529,5455,6740,8139,9826,
%U 11899,14299,16986,20615,24373,29023,34679,41447,49761,57395,68775,81535
%N Number of partitions p of n including round(mean(p)) as a part. (This is "Mathematica round")
%C Here, "round(x)" is "Round[x]" in Mathematica: round(x) = the integer nearest x if x is not of the form k + 1/2, where k is an integer, and round(k + 1/2) = the even integer nearest k. (Thus round(3/2) "rounds up" to 2, whereas round(5/2) "rounds down" to 2.) For round(x) defined as floor(x + 1/2), see A241733.
%F a(n) + A241339(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. A241334, A241339, A000041, A241312, A241733.
%K nonn,easy
%O 0,3
%A _Clark Kimberling_, Apr 20 2014
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