A103939 - OEIS

%I #19 Mar 29 2021 15:06:32

%S 1,1,3,8,32,136,722,3924,22954,138316,860364,5472444,35503288,

%T 234070648,1564945158,10589356592,72412611194,499788291616,

%U 3478059566250,24383023246284,172074483068320,1221654305104920,8720583728414354,62560709120463028,450854177292364660

%N Number of unrooted Eulerian 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 Andrew Howroyd, <a href="/A103939/b103939.txt">Table of n, a(n) for n = 0..500</a>

%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 For n > 0, a(n) = (1/(2n))*(2^n*binomial(2n, n)/(n+1) + Sum_{0<k<n, k|n} phi(n/k)*2^k*binomial(2k, k)) where phi is the Euler function A000010.

%t a[n_] := (1/(2n)) (2^n Binomial[2n, n]/(n+1) + Sum[Boole[0<k<n] EulerPhi[ n/k] 2^k Binomial[2k, k], {k, Divisors[n]}]);

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

%o (PARI) a(n)={if(n==0, 1, sumdiv(n, d, if(d<n, 1, 1/(n+1)) * eulerphi(n/d) * 2^d * binomial(2*d,d))/(2*n))} \\ _Andrew Howroyd_, Mar 29 2021

%Y Cf. A052701, A103940, A069727.

%K easy,nonn

%O 0,3

%A _Valery A. Liskovets_, Mar 17 2005

%E a(0)=1 prepended and terms a(21) and beyond from _Andrew Howroyd_, Mar 29 2021