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A241338
Number of partitions p of n including round(mean(p)) as a part. (This is "Mathematica round")
8
0, 1, 2, 3, 4, 5, 7, 10, 13, 18, 25, 31, 44, 57, 73, 99, 127, 166, 205, 268, 349, 424, 548, 674, 858, 1075, 1324, 1665, 1973, 2472, 3085, 3774, 4529, 5455, 6740, 8139, 9826, 11899, 14299, 16986, 20615, 24373, 29023, 34679, 41447, 49761, 57395, 68775, 81535
OFFSET
0,3
COMMENTS
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.
FORMULA
a(n) + A241339(n) = A000041(n) for n >= 0.
EXAMPLE
a(6) counts these 8 partitions: 6, 33, 321, 3111, 222, 2211, 21111, 111111.
MATHEMATICA
z = 30; f[n_] := f[n] = IntegerPartitions[n];
Table[Count[f[n], p_ /; MemberQ[p, Floor[Mean[p]]]], {n, 0, z}] (* A241334 *)
Table[Count[f[n], p_ /; ! MemberQ[p, Floor[Mean[p]]]], {n, 0, z}] (* A241335 *)
Table[Count[f[n], p_ /; MemberQ[p, Ceiling[Mean[p]]]], {n, 0, z}] (* A241336 *)
Table[Count[f[n], p_ /; ! MemberQ[p, Ceiling[Mean[p]]]], {n, 0, z}] (* A241337 *)
Table[Count[f[n], p_ /; MemberQ[p, Round[Mean[p]]]], {n, 0, z}] (* A241338 *)
Table[Count[f[n], p_ /; ! MemberQ[p, Round[Mean[p]]]], {n, 0, z}] (* A241339 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Apr 20 2014
STATUS
approved