OFFSET
0,3
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.
FORMULA
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.
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 *)
PROG
(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
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Valery A. Liskovets, Mar 17 2005
EXTENSIONS
More terms from Jean-François Alcover, Aug 30 2019
a(0)=1 prepended by Andrew Howroyd, Mar 29 2021
STATUS
approved