cp's OEIS Frontend

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).

%I A069732 #12 Aug 28 2019 03:57:41
%S A069732 0,0,2,7,40,239,1549,10396,71467,498598,3520015,25087426,180249182,
%T A069732 1304148015,9494015372,69492950976,511147940104,3776180492129,
%U A069732 28007532925171,208474866181148
%N 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).
%H A069732 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 A069732 a(n) = A069724(n) - A003645(n) - A069725(n).
%t A069732 A003645[n_] := 2^n CatalanNumber[n + 1];
%t A069732 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 A069732 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 A069732 a[n_] := If[n == 1, 0, A069724[n] - A003645[n - 2] - A069725[n]];
%t A069732 Array[a, 20] (* _Jean-François Alcover_, Aug 28 2019 *)
%Y A069732 Cf. A069724, A003645, A069725, A005470.
%K A069732 easy,nonn
%O A069732 1,3
%A A069732 _Valery A. Liskovets_, Apr 07 2002