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

1, 2, 4, 13, 50, 248, 1407, 8600, 55154, 365292, 2473956, 17053468, 119191992, 842688120, 6015275094, 43292026736, 313788095994, 2288506113056, 16781638172458, 123656774440396, 915123392599456

Offset: 0

Author

Valery A. Liskovets, Apr 07 2002

Keywords

• easy
• nonn

Comments

Unicursal (in a broad sense) means that no more than two vertices are of odd valency (that is, maps possessing an Eulerian path).

Links

• V. A. Liskovets and T. R. S. Walsh, Enumeration of Eulerian and unicursal planar maps, Discr. Math., 282 (2004), 209-221.

Crossrefs

Cf. A069727, A069724.

Programs

Mathematica

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]];
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;
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);
d[n_] := 4 - Mod[n, 2];
a[n_] := A069727[n] + A069724[n];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Aug 28 2019 *)

Formula

a(n) = A069727(n) + A069724(n).