login
Number of unrooted n-edge maps in the plane (planar with a distinguished outside face).
0

%I #12 Aug 28 2019 05:33:47

%S 2,6,26,150,1032,8074,67086,586752,5317226,49592424,473357994,

%T 4606116310,45554761836,456848968518,4637014782748,47563495004742,

%U 492422043299964,5140194991046122,54053208147441474,572191817441284272

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

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

%H V. A. Liskovets and T. R. Walsh, <a href="http://dx.doi.org/10.1016/j.aam.2005.03.006">Counting unrooted maps on the plane</a>, Advances in Applied Math., 36, No.4 (2006), 364-387.

%F 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.

%t 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];

%t q[n_] := If[EvenQ[n], 0, 3^((n-1)/2) Binomial[n-1, (n-1)/2]/(n+1)];

%t Array[a, 20] (* _Jean-François Alcover_, Aug 28 2019 *)

%Y Cf. A005159, A005470.

%K easy,nonn

%O 1,1

%A _Valery A. Liskovets_, Mar 17 2005