A069732 Number of nonisomorphic unrooted unicursal planar maps with n edges and no vertices of valency 1 (unicursal means that exactly two vertices are of odd valency; there is an Eulerian path).

0, 0, 2, 7, 40, 239, 1549, 10396, 71467, 498598, 3520015, 25087426, 180249182, 1304148015, 9494015372, 69492950976, 511147940104, 3776180492129, 28007532925171, 208474866181148

Offset

1,3

Links

Table of n, a(n) for n=1..20.

V. A. Liskovets and T. R. S. Walsh, Enumeration of Eulerian and unicursal planar maps, Discr. Math., 282 (2004), 209-221.

Formula

a(n) = A069724(n) - A003645(n) - A069725(n).

Mathematica

A003645[n_] := 2^n CatalanNumber[n + 1];
A069724[n_] := 1/(2 n) DivisorSum[n, If[OddQ[n/#], EulerPhi[n/#] 2^(# - 2) Binomial[2 #, #], 0] &] + If[OddQ[n], 2^((n - 3)/2) Binomial[n - 1, (n - 1)/2], 2^((n - 6)/2) Binomial[n, n/2]];
A069725[n_] := If[n <= 2, 1, With[{m = Floor[(n + 1)/2]}, 1/n 2^(n - 3) Binomial[2 n - 2, n - 1] + 2^(m - 3) Binomial[2 m - 2, m - 1]]];
a[n_] := If[n == 1, 0, A069724[n] - A003645[n - 2] - A069725[n]];
Array[a, 20] (* Jean-François Alcover, Aug 28 2019 *)

Crossrefs

Cf. A069724, A003645, A069725, A005470.

Sequence in context: A153744 A266422 A189826 * A346964 A277565 A157504

Adjacent sequences: A069729 A069730 A069731 * A069733 A069734 A069735

Keyword

easy,nonn

Author

Valery A. Liskovets, Apr 07 2002

Status

approved