A237754 Number of partitions of n such that 2*(greatest part) > (number of parts).

1, 1, 2, 4, 5, 8, 11, 16, 23, 32, 43, 59, 78, 104, 137, 181, 233, 303, 388, 497, 630, 799, 1003, 1262, 1574, 1961, 2430, 3008, 3701, 4551, 5569, 6805, 8284, 10070, 12195, 14753, 17786, 21413, 25709, 30824, 36856, 44014, 52435, 62384, 74062, 87811, 103901

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 not a part of p.

Examples

			a(5) = 5 counts these partitions:  5, 41, 32, 311, 221.
		

Links

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

Crossrefs

Cf. A064173, A237751-A237753, A237755-A237757, A000041.

Programs

Mathematica

z = 50; Table[Count[IntegerPartitions[n], p_ /; 2 Max[p] > Length[p]], {n, z}]

PARI

my(N=66, x='x+O('x^N)); Vec(sum(k=1, N, x^k*prod(j=1, k, (1-x^(2*k+j-2))/(1-x^j)))) \\ Seiichi Manyama, Jan 25 2022

Formula

a(n) = A000041(n) - A237752(n).

G.f.: Sum_{k>=1} x^k * Product_{j=1..k} (1-x^(2*k+j-2))/(1-x^j). - Seiichi Manyama, Jan 25 2022