A240219 Number of partitions p of n such that median(p) = mean(p).

1, 2, 3, 4, 4, 8, 5, 9, 10, 14, 7, 24, 8, 22, 31, 28, 10, 56, 11, 71, 68, 47, 13, 143, 69, 66, 147, 216, 16, 367, 17, 241, 304, 122, 509, 1019, 20, 163, 603, 1238, 22, 1712, 23, 1789, 3144, 286, 25, 3956, 1581, 2481, 2101, 4638, 28, 7739, 7357, 9209, 3737

Offset: 1

Author

Clark Kimberling, Apr 04 2014

Keywords

• nonn
• easy

Examples

			a(6) counts these 8 partitions:  6, 51, 42, 33, 331, 222, 2211, 111111.
		

Crossrefs

Cf. A240217, A240218, A240220, A240221, A000041.

Programs

Mathematica

z = 60; f[n_] := f[n] = IntegerPartitions[n];
Table[Count[f[n], p_ /; Median[p] < Mean[p]], {n, 1, z}]  (* A240217 *)
Table[Count[f[n], p_ /; Median[p] <= Mean[p]], {n, 1, z}] (* A240218 *)
Table[Count[f[n], p_ /; Median[p] == Mean[p]], {n, 1, z}] (* A240219 *)
Table[Count[f[n], p_ /; Median[p] > Mean[p]], {n, 1, z}]  (* A240220 *)
Table[Count[f[n], p_ /; Median[p] >= Mean[p]], {n, 1, z}] (* A240221 *)

Formula

a(n) = A240218(n) - A240217(n) for n >= 1.

a(n) + A240217(n) + A240220 = A000041(n) for n >= 1.