cp's OEIS Frontend

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

%I A103937 #18 Jan 22 2025 15:42:45
%S A103937 1,2,6,26,150,1032,8074,67086,586752,5317226,49592424,473357994,
%T A103937 4606116310,45554761836,456848968518,4637014782748,47563495004742,
%U A103937 492422043299964,5140194991046122,54053208147441474,572191817441284272,6093471300213162072,65245904156725935906
%N A103937 Number of unrooted n-edge maps in the plane (planar map with a distinguished outside face).
%D A103937 V. A. Liskovets and T. R. Walsh, Enumeration of unrooted maps on the plane, Rapport technique, UQAM, No. 2005-01, Montreal, Canada, 2005.
%H A103937 Andrew Howroyd, <a href="/A103937/b103937.txt">Table of n, a(n) for n = 0..500</a>
%H A103937 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 A103937 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 A103937 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 A103937 q[n_] := If[EvenQ[n], 0, 3^((n-1)/2) Binomial[n-1, (n-1)/2]/(n+1)];
%t A103937 Array[a, 20] (* _Jean-François Alcover_, Aug 28 2019 *)
%o A103937 (PARI) a(n) = {if(n==0, 1, (3^n*binomial(2*n,n)/(n+1) + sumdiv(n, k, if(k<n, eulerphi(n/k)*3^k*binomial(2*k,k))))/(2*n) + if(n%2, 3^((n-1)/2)*binomial(n-1,(n-1)/2)/(n+1)))} \\ _Andrew Howroyd_, Jan 22 2025
%Y A103937 Antidiagonal sums of A380240.
%Y A103937 Cf. A005159, A006384.
%K A103937 easy,nonn
%O A103937 0,2
%A A103937 _Valery A. Liskovets_, Mar 17 2005
%E A103937 a(0)=1 prepended by _Andrew Howroyd_, Jan 21 2025