OFFSET
0,3
COMMENTS
By duality, also the number of unrooted (sensed) isthmusless planar n-edge maps. An isthmus may also be called a bridge. - Andrew Howroyd, Mar 28 2021
REFERENCES
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
Andrew Howroyd, Table of n, a(n) for n = 0..500
V. A. Liskovets and T. R. S. Walsh, Counting Unrooted Loopless Planar Maps [Extended abstract]
V. A. Liskovets and T. R. S. Walsh, Counting unrooted loopless planar maps, Europ. J. Combin., 26:5 (2005), 651-663.
Timothy R. Walsh, Generating nonisomorphic maps without storing them, SIAM J. Algebraic Discrete Methods 4 (1983), no. 2, 161-178.
FORMULA
a(n) = (1/(2n))*[2(4n+1)*binomial(4n, n)/((n+1)*(3n+1)*(3n+2)) + Sum_{0<k<n, k|n}phi(n/k)*binomial(4k, k)+q(n)] where phi is the Euler function (A000010), q(n)=binomial(2n, (n-2)/2) if n is even and q(n)=2n*binomial(2n, (n-1)/2)/(n+1) if n is odd.
MATHEMATICA
a[n_] := If[n==0, 1, (1/(2n))(Sum[Binomial[4k, k] EulerPhi[n/k] Boole[ 0<k<n], {k, Divisors[n]}] + 2(4n+1) Binomial[4n, n]/((n+1)(3n+1)(3n+2)) + q[n])];
q[n_] := If[EvenQ[n], Binomial[2n, (n-2)/2], 2n Binomial[2n, (n-1)/2]/ (n+1)];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Aug 29 2019 *)
PROG
(PARI) a(n) = {if(n==0, 1, (sumdiv(n, d, if(d<n, 1, 2*(4*n+1)/((n+1)*(3*n+1)*(3*n+2))) * eulerphi(n/d) * binomial(4*d, d)) + if(n%2, 2*n/(n+1), 1)*binomial(2*n, (n-1)\2))/(2*n))} \\ Andrew Howroyd, Mar 28 2021
CROSSREFS
KEYWORD
nonn
AUTHOR
EXTENSIONS
More terms from Valery A. Liskovets, Dec 01 2003
a(17) and a(19) corrected by Sean A. Irvine, Mar 26 2017
STATUS
approved