A069732 - OEIS

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

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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