A103937 Number of unrooted n-edge maps in the plane (planar with a distinguished outside face).

2, 6, 26, 150, 1032, 8074, 67086, 586752, 5317226, 49592424, 473357994, 4606116310, 45554761836, 456848968518, 4637014782748, 47563495004742, 492422043299964, 5140194991046122, 54053208147441474, 572191817441284272

Offset

1,1

References

V. A. Liskovets and T. R. Walsh, Enumeration of unrooted maps on the plane, Rapport technique, UQAM, No. 2005-01, Montreal, Canada, 2005.

Links

Table of n, a(n) for n=1..20.

V. A. Liskovets and T. R. Walsh, Counting unrooted maps on the plane, Advances in Applied Math., 36, No.4 (2006), 364-387.

Formula

a(n)=(1/(2n))[3^n*binomial(2n, n)/(n+1) +sum_{0<k<n, k|n}phi(n/k)3^k*binomial(2k, k)]+q(n) where phi is the Euler function A000010, q(n)=0 if n is even and q(n)=3^((n-1)/2)binomial(n-1, (n-1)/2)/(n+1) if n is odd.

Mathematica

a[n_] := (1/(2n)) (3^n Binomial[2n, n]/(n+1) + Sum[Boole[0<k<n] EulerPhi[ n/k] 3^k Binomial[2k, k], {k, Divisors[n]}]) + q[n];
q[n_] := If[EvenQ[n], 0, 3^((n-1)/2) Binomial[n-1, (n-1)/2]/(n+1)];
Array[a, 20] (* Jean-François Alcover, Aug 28 2019 *)

Crossrefs

Cf. A005159, A005470.

Sequence in context: A052844 A247224 A052859 * A159311 A000629 A185994

Adjacent sequences: A103934 A103935 A103936 * A103938 A103939 A103940

Keyword

easy,nonn

Author

Valery A. Liskovets, Mar 17 2005

Status

approved