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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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

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

Table of n, a(n) for n=1..25.

V. A. Liskovets and T. R. Walsh, Counting unrooted maps on the plane, Advances in Applied Math., 36, No.4 (2006), 364-387.

FORMULA

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.

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 *)

CROSSREFS

Cf. A003645, A103939, A069727.

Sequence in context: A141494 A189843 A045612 * A162543 A039744 A319121

Adjacent sequences:  A103937 A103938 A103939 * A103941 A103942 A103943

KEYWORD

easy,nonn

AUTHOR

Valery A. Liskovets, Mar 17 2005

EXTENSIONS

More terms from Jean-François Alcover, Aug 30 2019

STATUS

approved

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Last modified March 7 10:38 EST 2021. Contains 341869 sequences. (Running on oeis4.)