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

1,4

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

Links

Table of n, a(n) for n=1..60.

Example

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

Mathematica

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

Prog

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

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.

Sequence in context: A075888 A075889 A181771 * A045766 A281668 A132817

Adjacent sequences: A238625 A238626 A238627 * A238629 A238630 A238631

Keyword

nonn,easy

Author

Clark Kimberling, Mar 02 2014

Status

approved