OFFSET
1,5
COMMENTS
Also, the number of partitions p of n such that (1/2)*max(p) is a part of p.
Also the number of even-length integer partitions of n with distinct middle parts. For example, the partition (4,3,2,1) has middle parts {2,3} so is counted under a(10), but (3,2,2,1) has middle parts {2,2} so is not counted under a(8). - Gus Wiseman, May 13 2023
LINKS
A. Blecher and A. Knopfmacher, Fixed points and matching points in partitions, Ramanujan J. 58 (2022), 23-41.
FORMULA
For all n, A027187(n) >= a(n). [Because when a partition of n has an odd number of parts, then it is not counted by this sequence (cf. A238478) and also some of the partitions with an even number of parts might be excluded here. Cf. Examples.] - Antti Karttunen, Feb 27 2014
From Jeremy Lovejoy, Sep 29 2022: (Start)
G.f.: Sum_{n>=1} q^(3*n)/Product_{k=1..2*n} (1-q^k).
a(n) ~ Pi/(2^(17/4)*3^(3/4)*n^(5/4))*exp(Pi*sqrt(2*n/3)). Proved by Blecher and Knopfmacher. (End)
EXAMPLE
a(6) counts these partitions: 51, 42, 2211 which all have an even number of parts, and their medians 3, 3 and 1.5 are not present. Note that the partitions 33 and 3111, although having an even number of parts, are not included in the count of a(6), but instead in that of A238478(6), as their medians, 3 for the former and 1 for the latter, are present in those partitions.
MATHEMATICA
Table[Count[IntegerPartitions[n], p_ /; !MemberQ[p, Median[p]]], {n, 40}]
(* also *)
Table[Count[IntegerPartitions[n], p_ /; MemberQ[p, Max[p]/2]], {n, 50}]
PROG
(Python)
from sympy.utilities.iterable import partitions
def A238479(n): return sum(1 for p in partitions(n) if (m:=max(p, default=0))&1^1 and m>>1 in p) # Chai Wah Wu, Sep 21 2023
CROSSREFS
These partitions have ranks A362617.
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Feb 27 2014
STATUS
approved