|
%I #13 Apr 29 2014 22:45:15
%S 1,0,0,0,1,2,4,5,9,12,17,25,33,44,62,77,104,131,180,222,278,368,454,
%T 581,717,883,1112,1345,1745,2093,2519,3068,3820,4688,5570,6744,8151,
%U 9738,11716,14199,16723,20210,24151,28582,33728,39373,48163,55979,65738
%N Number of partitions p of n not including round(mean(p)) as a part. (This is "Mathematica round"; for round(x) defined as floor(x + 1/2), see A241734.)
%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.)
%F a(n) + A241338(n) = A000041(n) for n >= 0.
%e a(6) counts these 4 partitions: 51, 42, 411, 3111.
%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, A241338, A000041, A241312, A241734.
%K nonn,easy
%O 0,6
%A _Clark Kimberling_, Apr 20 2014
|
