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A069725 Number of nonisomorphic unrooted unicursal planar maps with n edges and two vertices of valency 1 (unicursal means that exactly two vertices are of odd valency; there is an Eulerian path).

%I A069725 #12 Aug 28 2019 03:45:43
%S A069725 1,1,3,11,62,342,2152,13768,91800,622616,4301792,30100448,213019072,
%T A069725 1521473984,10954616064,79420280064,579300888960,4248201302400,
%U A069725 31302536066560,231638727063040
%N A069725 Number of nonisomorphic unrooted unicursal planar maps with n edges and two vertices of valency 1 (unicursal means that exactly two vertices are of odd valency; there is an Eulerian path).
%C A069725 There is an easy formula.
%H A069725 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.
%t A069725 a[1] = a[2] = 1;
%t A069725 a[n_] := With[{m = Floor[(n+1)/2]}, 1/n 2^(n-3) Binomial[2n-2, n-1] + 2^(m-3) Binomial[2m-2, m-1]];
%t A069725 Array[a, 20] (* _Jean-François Alcover_, Aug 28 2019 *)
%Y A069725 Cf. A069727, A003645, A069732.
%K A069725 easy,nonn
%O A069725 1,3
%A A069725 _Valery A. Liskovets_, Apr 07 2002