A396539 Number of bracelets (turnover necklaces) of length 4 using exactly 4 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.

3, 378, 14535, 583740, 20740020, 818890215, 33128023503, 1434342872340, 64413854851335, 3036649176072765, 147728685433584465, 7431492259342213440, 383693601452301792795, 20304027597295616632395, 1096964233375116271929753, 60410210457682687411896075, 3383028314439620964543868575

Offset

4,1

Comments

Let a(n) be the number of bracelets of length 4 using exactly 4 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 4 colors, there are 24 linear arrangements and the dihedral group D4 has 8 elements, giving 24/8 = 3 distinct bracelets per color set. Hence a(n) = 3 * binomial(N,4).

This sequence is the r=4 slice (m=4) of the 3D array A338859(4,n) = A321791(4, A000081(n)). Not to be confused with A214312 (varying length) or A214309 (representatives).

Links

Table of n, a(n) for n=4..20.

Formula

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

Example

For n=4, N = A000081(4) = 4 colors (A, B, C, D). The 3 bracelets are:
         (1) A B C D
         (2) A B D C
         (3) A C B D
All rotations and reflections are considered identical.

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(k):
    return 3 * comb(A000081(k), 4)
print([a(k) for k in range(4, 21)])

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).

Sequence in context: A068988 A136025 A370448 * A157577 A062604 A395836

Adjacent sequences: A396536 A396537 A396538 * A396540 A396541 A396542

Keyword

nonn

Author

Chen Wei-Shi, May 28 2026

Status

approved