A098692 Main diagonal of array in A098691.

1, 2, 10, 78, 777, 9800, 149796, 2690420, 55555500, 1296871224, 33773107758, 970753545580, 30527491279005, 1042604500906800, 38430716820193144, 1520662246114589640, 64291516462902839175, 2892426397164199846860, 137970526315789473684210, 6955460736173788715925048, 369510689788116404049535299

Offset: 1

Author

Ralf Stephan, Sep 21 2004

Keywords

• nonn

Comments

a(n) is the number of self-complementary n-bead primitive necklaces of n+1 colors (see Miller (1978)). - Petros Hadjicostas, Aug 21 2019

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..387
• H. Meyn and W. Götz, Self-reciprocal polynomials over finite fields, Séminaire Lotharingien de Combinatoire, B21d (1989), 8 pp.
• R. L. Miller, Necklaces, symmetries and self-reciprocal polynomials, Discr. Math. 22 (1978), 25-33.

Crossrefs

Cf. A098691.

Programs

Maple

with(numtheory):
a:= n-> `if`(n=2^ilog2(n) and n>1, (n+1)^n-1, add(mobius(d)*
       (n+1)^(n/d), d=select(x-> x::odd, divisors(n))))/(2*n):
seq(a(n), n=1..20);  # Alois P. Heinz, Aug 21 2019

Formula

a(n) = ((n + 1)^n - 1)/(2*n) if n = 2^s (for s >= 1), and (1/(2*n)) * Sum_{d|n, d odd} mu(d) * (n + 1)^(n/d) otherwise. - Petros Hadjicostas, Aug 21 2019

Extensions

More terms by Petros Hadjicostas, Aug 21 2019