cp's OEIS Frontend

a(n) is asymptotic to 2^(n+1)/n. More precisely, I conjecture for any m > 0, a(n) = 2^(n+1)/n * Sum_{k=0..m} A000670(k)/n^k + o(1/n^(m+1)) (A000670 = preferential arrangements of n labeled elements) which can be written a(n) = 2^n/n * 2 + Sum_{k=1..m} A000629(k)/n^k + o(1/n^(m+1)) (A000629 = necklaces of sets of labeled beads). In fact I conjecture that a(n) = 2^(n+1)/n * (1 + 1/n + 3/n^2 + 13/n^3 + 75/n^4 + 541/n^5 + o(1/n^5)). - Benoit Cloitre, Oct 20 2002

A082550(n) = a(n+1) - a(n). - Reinhard Zumkeller, Feb 19 2006

This sequence and A325347 partition the partition numbers, A000041.

The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length). - Gus Wiseman, Mar 16 2023

The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).

Also a(n+1) = number of distinct (infinite) output sequences from binary n-stage shift register which feeds back the complement of the sum of its contents. E.g., for n=5 there are 6 such sequences.

Also a(n+1) = number of binary vectors (x_1,...x_n) satisfying Sum_{i=1..n} i*x_i = 0 (mod n+1) = size of Varshamov-Tenengolts code VT_0(n). E.g., |VT_0(5)| = 6 = a(6).

Number of binary necklaces with an odd number of zeros. - Joerg Arndt, Oct 26 2015

Also, number of subsets of {1,2,...,n-1} which sum to 0 modulo n (cf. A063776). - Max Alekseyev, Mar 26 2016

From Gus Wiseman, Sep 14 2019: (Start)

Also the number of subsets of {1..n} containing n whose mean is an element. For example, the a(1) = 1 through a(8) = 16 subsets are:

1 2 3 4 5 6 7 8

123 234 135 246 147 258

345 456 357 468

12345 1236 567 678

1456 2347 1348

23456 2567 1568

12467 3458

13457 3678

34567 12458

1234567 14578

23578

24568

45678

123468

135678

2345678

(End)

Number of self-dual binary necklaces with 2n beads (cf. A263768, A007147). - Bernd Mulansky, Apr 25 2023

Note that such a partition cannot be strict for n > 1.

Conjecture: If n is squarefree, then a(n) = 0.

From Gus Wiseman, Sep 14 2019: (Start)

Also the number of subsets of {1..n} that are empty or contain n and have integer mean. If the subsets are not required to contain n, we get A327475. For example, the a(1) = 2 through a(6) = 12 subsets are:

{} {} {} {} {} {}

{1} {2} {3} {4} {5} {6}

{1,3} {2,4} {1,5} {2,6}

{1,2,3} {2,3,4} {3,5} {4,6}

{1,3,5} {1,2,6}

{3,4,5} {1,5,6}

{1,2,4,5} {2,4,6}

{1,2,3,4,5} {4,5,6}

{1,2,3,6}

{1,4,5,6}

{2,3,5,6}

{2,3,4,5,6}

(End)

The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).