A069732 - OEIS

%I #12 Aug 28 2019 03:57:41

%S 0,0,2,7,40,239,1549,10396,71467,498598,3520015,25087426,180249182,

%T 1304148015,9494015372,69492950976,511147940104,3776180492129,

%U 28007532925171,208474866181148

%N 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).

%H V. A. Liskovets and T. R. S. Walsh, <a href="https://doi.org/10.1016/j.disc.2003.09.015">Enumeration of Eulerian and unicursal planar maps</a>, Discr. Math., 282 (2004), 209-221.

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

%t A003645[n_] := 2^n CatalanNumber[n + 1];

%t 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]];

%t 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]]];

%t a[n_] := If[n == 1, 0, A069724[n] - A003645[n - 2] - A069725[n]];

%t Array[a, 20] (* _Jean-François Alcover_, Aug 28 2019 *)

%Y Cf. A069724, A003645, A069725, A005470.

%K easy,nonn

%O 1,3

%A _Valery A. Liskovets_, Apr 07 2002