OFFSET
0,3
COMMENTS
By duality, also the number of unrooted (sensed) bipartite maps with n edges. - Andrew Howroyd, Mar 29 2021
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..1000
M. Deryagina, On the enumeration of hypermaps which are self-equivalent with respect to reversing the colors of vertices, Preprint 2016.
V. A. Liskovets and T. R. S. Walsh, Enumeration of Eulerian and unicursal planar maps, Discr. Math., 282 (2004), 209-221.
FORMULA
a(n) = (1/(2n))*(3*2^(n-1)*binomial(2n, n)/((n+1)(n+2)) + Sum_{k=1..n-1, k|n} phi(n/k)*d(n/k)*2^(k-2)*binomial(2k, k)) + q(n) where phi is the Euler function A000010, q(n) = 2^((n-4)/2)*binomial(n, n/2)/(n+2) if n is even, q(n) = 2^((n-1)/2)*binomial(n-1, (n-1)/2)/(n+1) if n is odd, d(n)=4, if n is even and d(n)=3 if n is odd. - Valery A. Liskovets, Mar 17 2005
a(n) ~ 3 * 2^(3*n-2) / (sqrt(Pi) * n^(7/2)). - Vaclav Kotesovec, Aug 28 2019
MATHEMATICA
a[n_] := (1/(2n)) * (3*2^(n-1) * Binomial[2n, n]/((n+1)*(n+2)) + Sum[ EulerPhi[n/k] * d[n/k] * 2^(k-2) * Binomial[2k, k], {k, Most[ Divisors[n]]}]) + q[n]; a[0] = 1; q[n_?EvenQ] := 2^((n-4)/2)*Binomial[ n, n/2]/(n+2); q[n_?OddQ] := 2^((n-1)/2)*Binomial[(n-1), (n-1)/2]/(n+1); d[n_] := 4-Mod[n, 2]; Table[ a[n], {n, 0, 20}] (* Jean-François Alcover, Dec 19 2011, after Valery A. Liskovets *)
PROG
(PARI) a(n) = {if(n==0, 1, sumdiv(n, d, if(d<n, 1, 2/((n+1)*(n+2))) * eulerphi(n/d) * (4-n/d%2) * 2^(d-2) * binomial(2*d, d))/(2*n) + if(n%2, 2^((n-1)/2)/(n+1), 2^((n-4)/2)/(n+2))*binomial(n\2*2, n\2))} \\ Andrew Howroyd, Mar 29 2021
CROSSREFS
KEYWORD
easy,nice,nonn
AUTHOR
Valery A. Liskovets, Apr 07 2002
STATUS
approved