A113183 Number of unrooted two-face maps in the plane (considered up to orientation-preserving homeomorphism) with the faces of equal degree n: planar maps with a distinguished outside face.

1, 1, 2, 3, 8, 18, 58, 155, 546, 1592, 5774, 17798, 65676, 210362, 785248, 2588155, 9743348, 32832290, 124416022, 426685544, 1625465732, 5654938190, 21636274202, 76171463926, 292498386900, 1040120036300, 4006388161846, 14369121494126

Offset: 1

Author

Valery A. Liskovets, Oct 19 2005

Keywords

• nonn

Examples

			There exist 2 maps in the plane with two triangular faces: a triangle and a map consisting of a 2-path and a loop in its middle vertex that separates both ends. Therefore a(3) = 2.
		

Links

• M. Bousquet, G. Labelle and P. Leroux, Enumeration of planar two-face maps, Discrete Math., vol. 222 (2000), 1-25.

Crossrefs

Cf. A000010, A060404, A113181.

Programs

Mathematica

a[n_] := DivisorSum[n, EulerPhi[#] * Binomial[n/# - 1, Floor[n/(2*#)]]^2 &] / n; Array[a, 30] (* Amiram Eldar, Aug 24 2023 *)

PARI

a(n) = sumdiv(n, k, eulerphi(k)*binomial(n/k - 1, n\(2*k))^2)/n; \\ Michel Marcus, Oct 14 2015

Formula

a(n) = (1/n) Sum_{k|n} phi(k) C((n/k)-1,floor(n/(2k)))^2 where phi(k) is the Euler function A000010.