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A103941 Number of unrooted loopless n-edge maps in the plane (planar with a distinguished outside face).

%I A103941 #19 Mar 28 2021 19:32:05
%S A103941 1,1,2,6,22,103,614,3872,26414,186988,1367976,10254326,78461338,
%T A103941 610598818,4821248244,38546510368,311560875422,2542507084588,
%U A103941 20925300483992,173530381632724,1448900079476152,12172334379246523,102833593763830038,873187910184763024,7449120536014301138
%N A103941 Number of unrooted loopless n-edge maps in the plane (planar with a distinguished outside face).
%D A103941 V. A. Liskovets and T. R. Walsh, Enumeration of unrooted maps on the plane, Rapport technique, UQAM, No. 2005-01, Montreal, Canada, 2005.
%H A103941 Andrew Howroyd, <a href="/A103941/b103941.txt">Table of n, a(n) for n = 0..500</a>
%H A103941 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 A103941 For n > 0, a(n) = (1/(2n))*[binomial(4n, n)/(3n+1) + Sum_{0<k<n, k|n} phi(n/k)*binomial(4k, k)+q(n)] where phi is the Euler function A000010, q(n)=0 if n is even and q(n)=binomial(2n, (n-1)/2) if n is odd.
%t A103941 a[n_] := (1/(2n)) (Binomial[4n, n]/(3n+1) + Sum[Boole[0 < k < n] EulerPhi[ n/k] Binomial[4k, k], {k, Divisors[n]}] + q[n]);
%t A103941 q[n_] := If[EvenQ[n], 0, Binomial[2n, (n-1)/2]];
%t A103941 Array[a, 20] (* _Jean-François Alcover_, Sep 01 2019 *)
%o A103941 (PARI) a(n) = {if(n==0, 1, (sumdiv(n, d, if(d<n, 1, 1/(3*n+1)) * eulerphi(n/d) * binomial(4*d,d)) + if(n%2, binomial(2*n, (n-1)/2)))/(2*n))} \\ _Andrew Howroyd_, Mar 28 2021
%Y A103941 Cf. A002293, A006390, A103942, A000260, A005470.
%K A103941 easy,nonn
%O A103941 0,3
%A A103941 _Valery A. Liskovets_, Mar 17 2005
%E A103941 a(0)=1 prepended and terms a(21) and beyond from _Andrew Howroyd_, Mar 28 2021