A237752 Number of partitions of n such that 2*(greatest part) <= (number of parts).

0, 1, 1, 1, 2, 3, 4, 6, 7, 10, 13, 18, 23, 31, 39, 50, 64, 82, 102, 130, 162, 203, 252, 313, 384, 475, 580, 710, 864, 1053, 1273, 1544, 1859, 2240, 2688, 3224, 3851, 4602, 5476, 6514, 7727, 9160, 10826, 12791, 15072, 17747, 20853, 24481, 28679, 33577, 39231

Offset: 1

Author

Clark Kimberling, Feb 13 2014

Keywords

• nonn
• easy

Comments

Also, the number of partitions of n such that (greatest part) >= 2*(number of parts); hence, the number of partitions of n such that (rank + greatest part) <= 0.

Also, the number of partitions p of n such that max(max(p), 2*(number of parts of p)) is a part of p.

Examples

			The partitions of 6 that do not qualify are 22311, 21111, 111111, so that a(6) = 11 - 3 = 8.
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A064173, A237751, A237753-A237757, A000041.

Programs

Mathematica

z = 50; Table[Count[IntegerPartitions[n], p_ /; 2 Max[p] <= Length[p]], {n, z}]
(* also *)
Table[Count[IntegerPartitions[n], p_ /; MemberQ[p, Max[Max[p],2*Length[p]]]], {n, 50}]

Formula

a(n) = A000041(n) - A237754(n).