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A386388 a(n) is the number of complement pairs of 2n-bead balanced bicolor necklaces.

%I A386388 #34 Jul 30 2025 23:36:22
%S A386388 0,0,0,1,3,11,36,118,395,1337,4598,15986,56270,199854,716132,2584754,
%T A386388 9391051,34315811,126040590,465062362,1723070794,6407806952,
%U A386388 23910175804,89493721076,335912391966,1264105728842,4768446886764,18027215662284,68291878325138
%N A386388 a(n) is the number of complement pairs of 2n-bead balanced bicolor necklaces.
%C A386388 A003239(n) is the number of 2n-bead balanced bicolor necklaces, and A000013(n) is the number of those that are self-complementary (i.e., can be rotated so that all beads change color). Their difference 2*a(n) is the number of those that are not self-complementary. a(n) is the number pairs of distinct complements.
%C A386388 Doubled entries: 0, 0, 0, 2, 6, 22, 72, 236, 790, 2674, 9196, 31972, 112540, 399708, 1432264, ...
%H A386388 Michael De Vlieger, <a href="/A386388/b386388.txt">Table of n, a(n) for n = 0..1670</a>
%F A386388 a(n) = (A003239(n) - A000013(n)) / 2.
%e A386388   n  | A003239(n) A000013(n) | 2*a(n)      a(n)
%e A386388   0  |         1          1  |     0         0
%e A386388   1  |         1          1  |     0         0
%e A386388   2  |         2          2  |     0         0
%e A386388   3  |         4          2  |     2         1
%e A386388   4  |        10          4  |     6         3
%e A386388   5  |        26          4  |    22        11
%e A386388   6  |        80          8  |    72        36
%e A386388   7  |       246         10  |   236       118
%e A386388   8  |       810         20  |   790       395
%e A386388   9  |      2704         30  |  2674      1337
%e A386388  10  |      9252         56  |  9196      4598
%e A386388 Examples for n=4 with necklaces of length 8:
%e A386388 A000013(4) = 4 necklaces are self-complementary:
%e A386388  00001111, 00110011, 01010101, 00101101 (compare A385665)
%e A386388 There are a(n) = 3 pairs of complementary necklaces:
%e A386388  (00110101, 00101011), (00100111, 00011011), (00010111, 00011101)
%t A386388 a[0]=0;a[n_]:=( Sum[ EulerPhi[n/k]*Binomial[2k, k]/(2n), {k, Divisors[n]}]- Fold[ #1 + EulerPhi[2#2]2^(n/#2)/(2n) &, 0, Divisors[n]])/2;Array[a,29,0] (* _James C. McMahon_, Jul 30 2025 *)
%Y A386388 Cf. A000013, A003239, A385665.
%K A386388 nonn
%O A386388 0,5
%A A386388 _Tilman Piesk_, Jul 20 2025