A072506 Triangle giving T(n,m) = number of necklaces of two colors with 2n beads of which m=1..n are black.

1, 1, 2, 1, 3, 4, 1, 4, 7, 10, 1, 5, 12, 22, 26, 1, 6, 19, 43, 66, 80, 1, 7, 26, 73, 143, 217, 246, 1, 8, 35, 116, 273, 504, 715, 810, 1, 9, 46, 172, 476, 1038, 1768, 2438, 2704, 1, 10, 57, 245, 776, 1944, 3876, 6310, 8398, 9252, 1, 11, 70, 335, 1197, 3399, 7752

Offset: 1

Author

Wouter Meeussen, Aug 03 2002

Keywords

• nonn
• tabl

Comments

Left half of even rows of triangle A047996 (with the leftmost edge discarded).

Crossrefs

Penultimate entries give binary necklaces of n-1 black beads and n+1 white beads, presumably A007595, antepenultimate entries give binary necklaces of n-2 black beads and n+2 white beads, presumably A003444.

Cf. A047996, A003444.

Programs

Mathematica

Table[(Plus@@(EulerPhi[ # ]Binomial[2n/#, m/# ] &)/@Intersection[Divisors[2n], Divisors[m]])/(2n), {n, 13}, {m, n}]

Formula

(1/(2n)) Sum_{d |(2n, m)} phi(d)*binomial(2n/d, m/d)