0,4

Number of partitions whose summands can be divided into two sets whose sums are equal, or whose sums differ by one.

Alois P. Heinz, Table of n, a(n) for n = 0..115 (first 77 terms from Chai Wah Wu)

a(n) = A000041(n) - A046663(n,floor(n/2)). - Max Alekseyev, Oct 17 2022

For example: the partition of 6 into 2+2+2 is not counted because no subset of {2,2,2} has the sum 3. The partition of 6 into 3+2+1 is counted because {3,2,1} can be divided into two subsets {3} and {2,1} each of which has sum 3. The partition of 7 into 4+2+1 is counted because {4,2,1} can be broken up into sets {4} and {2,1} the sum of whose elements differ by one.

N:= 50: # to get a(0) .. a(N)

Pm1:= combinat:-partition(0);

for m from 0 to N/2 do

Pm:= Pm1;

A[2*m]:= nops({seq(seq(sort([op(Pm[i]), op(Pm[j])]), j=1..i), i=1..nops(Pm))});

if 2*m+1 > N then break fi;

Pm1:= combinat:-partition(m+1);

A[2*m+1]:= nops({seq(seq(sort([op(Pm[i]), op(Pm1[j])]), j=1..nops(Pm1)), i=1..nops(Pm))});

od:

seq(A[i], i= 0..N); # Robert Israel, Aug 18 2016

(Python)

from collections import Counter

from sympy.utilities.iterables import partitions

def A276107(n): return len({tuple(sorted((p+q).items())) for p in (Counter(p) for p in partitions(n>>1)) for q in (Counter(q) for q in partitions(n+1>>1))}) # Chai Wah Wu, Sep 20 2023

Cf. A000041, A002219, A046663.

Sequence in context: A240490 A039839 A039844 * A187068 A187070 A038196

Adjacent sequences: A276104 A276105 A276106 * A276108 A276109 A276110

David S. Newman, Aug 18 2016

a(11)-a(47) from Alois P. Heinz, Aug 18 2016

approved