A241338 Number of partitions p of n including round(mean(p)) as a part. (This is "Mathematica round")

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.

Links

Table of n, a(n) for n=0..48.

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

Cf. A241334, A241339, A000041, A241312, A241733.

Sequence in context: A160333 A174578 A241733 * A271489 A018127 A017835

Adjacent sequences: A241335 A241336 A241337 * A241339 A241340 A241341

Keyword

nonn,easy

Author

Clark Kimberling, Apr 20 2014

Status

approved