A238628 Number of partitions p of n such that n - max(p) is a part of p.

0, 1, 1, 3, 2, 5, 3, 8, 4, 11, 5, 16, 6, 21, 7, 29, 8, 38, 9, 51, 10, 66, 11, 88, 12, 113, 13, 148, 14, 190, 15, 246, 16, 313, 17, 402, 18, 508, 19, 646, 20, 812, 21, 1023, 22, 1277, 23, 1598, 24, 1982, 25, 2461, 26, 3036, 27, 3745, 28, 4593, 29, 5633

Offset: 1

Author

Clark Kimberling, Mar 02 2014

Keywords

• nonn
• easy

Comments

Also the number of integer partitions of n that are of length 2 or contain n/2. The first condition alone is A004526, complement A058984. The second condition alone is A035363, complement A086543, ranks A344415. - Gus Wiseman, Oct 07 2023

Examples

			a(6) counts these partitions:  51, 42, 33, 321, 3111.
		

Crossrefs

Cf. A238479.

The strict case is A365659, complement A365826.

The complement is counted by A365825.

These partitions are ranked by A366318.

A000041 counts integer partitions, strict A000009.

A140106 counts strict partitions of length 2, complement A365827.

A182616 counts partitions of 2n that do not contain n, strict A365828.

Cf. A004526, A005408, A008967, A035363, A058984, A068911, A086543.

Programs

Mathematica

Table[Count[IntegerPartitions[n], p_ /; MemberQ[p, n - Max[p]]], {n, 50}]

PARI

a(n) = my(res = floor(n/2)); if(!bitand(n, 1), res+=(numbpart(n/2)-1)); res

Python

from sympy.utilities.iterables import partitions
def A238628(n): return sum(1 for p in partitions(n) if n-max(p,default=0) in p) # Chai Wah Wu, Sep 21 2023