A103941 Number of unrooted loopless n-edge maps in the plane (planar with a distinguished outside face).

1, 1, 2, 6, 22, 103, 614, 3872, 26414, 186988, 1367976, 10254326, 78461338, 610598818, 4821248244, 38546510368, 311560875422, 2542507084588, 20925300483992, 173530381632724, 1448900079476152, 12172334379246523, 102833593763830038, 873187910184763024, 7449120536014301138

Offset: 0

Author

Valery A. Liskovets, Mar 17 2005

Keywords

• easy
• nonn

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.

Crossrefs

Cf. A002293, A006390, A103942, A000260, A005470.

Programs

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 *)

PARI

a(n) = {if(n==0, 1, (sumdiv(n, d, if(dAndrew Howroyd, Mar 28 2021

Formula

For n > 0, a(n) = (1/(2n))*[binomial(4n, n)/(3n+1) + Sum_{0A000010, q(n)=0 if n is even and q(n)=binomial(2n, (n-1)/2) if n is odd.

Extensions

a(0)=1 prepended and terms a(21) and beyond from Andrew Howroyd, Mar 28 2021