A103940 Number of unrooted bipartite n-edge maps in the plane (planar with a distinguished outside face).

1, 1, 2, 5, 18, 72, 368, 1982, 11514, 69270, 430384, 2736894, 17752884, 117039548, 782480424, 5294705752, 36206357114, 249894328848, 1739030128872, 12191512867814, 86037243899240, 610827161152012, 4360291880624504, 31280354620428378, 225427088761560916, 1631398499577667252

Offset: 0

Author

Valery A. Liskovets, Mar 17 2005

Keywords

• easy
• nonn

Comments

Bipartite planar maps are dual to Eulerian planar maps.

References

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

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..500
• V. A. Liskovets and T. R. Walsh, Counting unrooted maps on the plane, Advances in Applied Math., 36, No.4 (2006), 364-387.

Crossrefs

Cf. A003645, A103939, A069727.

Programs

Mathematica

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];
d[n_] := If[EvenQ[n], 2, 1];
q[n_] := If[EvenQ[n], 0, 2^((n-1)/2) Binomial[n-1, (n-1)/2]/(n+1)];
Array[a, 25] (* Jean-François Alcover, Aug 30 2019 *)

PARI

a(n)={if(n==0, 1, sumdiv(n, d, if(dAndrew Howroyd, Mar 29 2021

Formula

For n > 0, a(n) = (1/(2n))*[2^(n-1)*binomial(2n, n)/(n+1) + Sum_{0A000010, 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.

Extensions

More terms from Jean-François Alcover, Aug 30 2019

a(0)=1 prepended by Andrew Howroyd, Mar 29 2021