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%I #19 Mar 29 2021 15:06:40
%S 1,1,2,5,18,72,368,1982,11514,69270,430384,2736894,17752884,117039548,
%T 782480424,5294705752,36206357114,249894328848,1739030128872,
%U 12191512867814,86037243899240,610827161152012,4360291880624504,31280354620428378,225427088761560916,1631398499577667252
%N Number of unrooted bipartite n-edge maps in the plane (planar with a distinguished outside face).
%C Bipartite planar maps are dual to Eulerian planar maps.
%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="/A103940/b103940.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-1)*binomial(2n, n)/(n+1) + Sum_{0<k<n, k|n} phi(n/k)*d(n/k)*2^(k-1)*binomial(2k, k)] + q(n) where phi is the Euler function A000010, d(n)=2, q(n)=0 if n is even and d(n)=1, q(n)=2^((n-1)/2)*binomial(n-1, (n-1)/2)/(n+1) if n is odd.
%t a[n_] := (1/(2 n)) (2^(n - 1) Binomial[2 n, n]/(n+1) + Sum[Boole[0 < k < n] EulerPhi[n/k] d[n/k] 2^(k-1) Binomial[2k, k], {k, Divisors[n]}]) + q[n];
%t d[n_] := If[EvenQ[n], 2, 1];
%t q[n_] := If[EvenQ[n], 0, 2^((n-1)/2) Binomial[n-1, (n-1)/2]/(n+1)];
%t Array[a, 25] (* _Jean-François Alcover_, Aug 30 2019 *)
%o (PARI) a(n)={if(n==0, 1, sumdiv(n, d, if(d<n, 1, 1/(n+1)) * eulerphi(n/d) * (2-n/d%2) * 2^(d-1) * binomial(2*d,d))/(2*n) + if(n%2, 2^((n-1)/2)*binomial(n-1,(n-1)/2)/(n+1)))} \\ _Andrew Howroyd_, Mar 29 2021
%Y Cf. A003645, A103939, A069727.
%K easy,nonn
%O 0,3
%A _Valery A. Liskovets_, Mar 17 2005
%E More terms from _Jean-François Alcover_, Aug 30 2019
%E a(0)=1 prepended by _Andrew Howroyd_, Mar 29 2021
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