A396566 Number of bracelets (turnover necklaces) of length 4 using exactly 3 colors, where the colors are chosen from a palette of size N = A000081(n) (the number of unlabeled rooted trees with n nodes), and each color corresponds to a distinct tree shape.

24, 504, 6840, 103776, 1481430, 23148840, 370145514, 6239664480, 108190392360, 1946102171640, 35845601561076, 677067443453190, 13040933356636620, 255861630224692890, 5098732402525355154, 103074299201812858200, 2110069526732687732856, 43695406785298898577360, 914231452074242154776820

Offset

4,1

Comments

Let a(n) be the number of bracelets of length 4 using exactly 3 colors, where the colors are chosen from a set of size N = A000081(n) (unlabeled rooted trees with n nodes). For a fixed set of 3 colors, the color distribution must be (2,1,1). There are 3 choices for the color that appears twice, and for each such choice, there are 2 distinct bracelets (the two occurrences can be adjacent or opposite). Hence there are 3 * 2 = 6 distinct bracelets per color set. Therefore a(n) = 6 * binomial(N,3). Offset 4,1 because n >= 4.

This sequence is the r=3 slice (m=4) of the same 3D array as A396539 (r=4). Not to be confused with A214310 (varying length) or A214307 (representatives).

Links

Table of n, a(n) for n=4..22.

Formula

a(n) = 6 * binomial(A000081(n), 3) for n >= 4.

Example

For n=4, N = A000081(4) = 4 colors (A, B, C, D). Choosing colors A, B, C with A appearing twice, the 2 bracelets are:
(1) A B A C
(2) A B C A
All rotations and reflections are considered identical. For n=5, N = 9 colors, there are 504 distinct bracelets.

Prog

(Python)
from math import comb
from functools import lru_cache
@lru_cache(maxsize=None)
def A000081(n):
    if n <= 1:
        return 1
    total = 0
    for j in range(1, n):
        s = 0
        for d in range(1, j + 1):
            if j % d == 0:
                s += d * A000081(d)
        total += s * A000081(n - j)
    return total // (n - 1)
def a(n):
    return 6 * comb(A000081(n), 3)
print([a(n) for n in range(4, 23)])

Crossrefs

Cf. A000081 (number of unlabeled rooted trees with n nodes, gives total colors N), A321791 (bracelets of length n using up to k colors), A338859 (unicyclic graphs with m cycle nodes and trees of size k), A396539 (r=4 slice of the same 3D array).

Sequence in context: A112390 A166757 A289833 * A242524 A264956 A383196

Adjacent sequences: A396562 A396563 A396564 * A396567 A396568 A396569

Keyword

nonn

Author

Chen Wei-Shi, May 29 2026

Status

approved