OFFSET
0,3
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))*[binomial(4n, n)/(3n+1) + Sum_{0<k<n, k|n} phi(n/k)*binomial(4k, k)+q(n)] where phi is the Euler function A000010, q(n)=0 if n is even and q(n)=binomial(2n, (n-1)/2) if n is odd.
MATHEMATICA
a[n_] := (1/(2n)) (Binomial[4n, n]/(3n+1) + Sum[Boole[0 < k < n] EulerPhi[ n/k] Binomial[4k, k], {k, Divisors[n]}] + q[n]);
q[n_] := If[EvenQ[n], 0, Binomial[2n, (n-1)/2]];
Array[a, 20] (* Jean-François Alcover, Sep 01 2019 *)
PROG
(PARI) a(n) = {if(n==0, 1, (sumdiv(n, d, if(d<n, 1, 1/(3*n+1)) * eulerphi(n/d) * binomial(4*d, d)) + if(n%2, binomial(2*n, (n-1)/2)))/(2*n))} \\ Andrew Howroyd, Mar 28 2021
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Valery A. Liskovets, Mar 17 2005
EXTENSIONS
a(0)=1 prepended and terms a(21) and beyond from Andrew Howroyd, Mar 28 2021
STATUS
approved