A240204 Number of partitions p of n such that mean(p) <= multiplicity(min(p)).

0, 1, 1, 1, 3, 3, 6, 6, 11, 14, 20, 25, 38, 45, 64, 85, 108, 140, 190, 227, 303, 387, 473, 606, 785, 926, 1183, 1496, 1816, 2208, 2778, 3345, 4170, 4990, 6031, 7424, 9097, 10558, 12926, 15750, 18900, 21987, 26660, 31838, 38392, 44798, 52731, 63184, 75620

Offset: 0

Author

Clark Kimberling, Apr 03 2014

Keywords

• nonn
• easy

Examples

			a(6) counts these 5 partitions:  411, 3111, 222, 2211, 21111, 111111.
		

Crossrefs

Cf. A240203, A240205, A240206, A240079, A000041.

Programs

Mathematica

z = 60; f[n_] := f[n] = IntegerPartitions[n];
t1 = Table[Count[f[n], p_ /; Mean[p] < Count[p, Min[p]]], {n, 0, z}]  (* A240203 *)
t2 = Table[Count[f[n], p_ /; Mean[p] <= Count[p, Min[p]]], {n, 0, z}] (* A240204 *)
t3 = Table[Count[f[n], p_ /; Mean[p] == Count[p, Min[p]]], {n, 0, z}] (* A240205 *)
t4 = Table[Count[f[n], p_ /; Mean[p] > Count[p, Min[p]]], {n, 0, z}] (* A240206 *)
t5 = Table[Count[f[n], p_ /; Mean[p] >= Count[p, Min[p]]], {n, 0, z}] (* A240079 *)

Formula

a(n) = A240203(n) + A240205(n) for n >= 0.

a(n) + A240206(n) = A000041(n) for n >= 0.