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A069730 Number of nonisomorphic unrooted unicursal planar maps with n edges.

%I A069730 #16 Aug 28 2019 09:56:39
%S A069730 1,2,4,13,50,248,1407,8600,55154,365292,2473956,17053468,119191992,
%T A069730 842688120,6015275094,43292026736,313788095994,2288506113056,
%U A069730 16781638172458,123656774440396,915123392599456
%N A069730 Number of nonisomorphic unrooted unicursal planar maps with n edges.
%C A069730 Unicursal (in a broad sense) means that no more than two vertices are of odd valency (that is, maps possessing an Eulerian path).
%H A069730 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 A069730 a(n) = A069727(n) + A069724(n).
%t A069730 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 A069730 A069727[n_] := (1/(2 n))*(3*2^(n - 1)*Binomial[2 n, n]/((n + 1)*(n + 2)) + Sum[EulerPhi[n/k]*d[n/k]*2^(k - 2)*Binomial[2 k, k], {k, Most[Divisors[n]]}]) + q[n]; A069727[0] = 1;
%t A069730 q[n_?EvenQ] := 2^((n - 4)/2)*Binomial[n, n/2]/(n + 2); q[n_?OddQ] := 2^((n - 1)/2)*Binomial[(n - 1), (n - 1)/2]/(n + 1);
%t A069730 d[n_] := 4 - Mod[n, 2];
%t A069730 a[n_] := A069727[n] + A069724[n];
%t A069730 Table[a[n], {n, 0, 20}] (* _Jean-François Alcover_, Aug 28 2019 *)
%Y A069730 Cf. A069727, A069724.
%K A069730 easy,nonn
%O A069730 0,2
%A A069730 _Valery A. Liskovets_, Apr 07 2002