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

1,3

Links

Table of n, a(n) for n=1..28.

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

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.

Example

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.

Mathematica

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

Prog

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

Crossrefs

Cf. A000010, A060404, A113181.

Sequence in context: A158448 A073192 A317722 * A157015 A240645 A273754

Adjacent sequences: A113180 A113181 A113182 * A113184 A113185 A113186

Keyword

nonn

Author

Valery A. Liskovets, Oct 19 2005

Status

approved