A103943 Number of unrooted two-vertex n-edge maps in the plane (planar with a distinguished outside face).

1, 3, 12, 48, 196, 798, 3248, 13184, 53416, 216018, 872344, 3518496, 14177528, 57080572, 229657792, 923474944, 3711572176, 14911097514, 59883185096, 240416320928, 964947251544, 3872021946532, 15533828715232, 62306843932928

Offset: 1

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

• G. C. Greubel, Table of n, a(n) for n = 1..1000
• 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. A060404, A033504.

Programs

Mathematica

f[n_] := (2^(2n - 1) - Binomial[2n - 1, n - 1] + Binomial[n - 1, Floor[n/2]])/2; Table[ f[n], {n, 24}] (* Robert G. Wilson v, Mar 24 2005 *)
Rest[CoefficientList[Series[1/8(-2+2/(1-4x)-1/Sqrt[1-4x]+1/Sqrt[1+4x]+2/Sqrt[-1+2/(1+2x)]-Sqrt[1+Sqrt[1-16x^2]]/Sqrt[1/2-8x^2]), {x, 0, 20}], x]] (* Benedict W. J. Irwin, Aug 13 2016 *)

Formula

2a(n) = 2^(2n-1) - binomial(2n-1, n-1) + binomial(n-1, floor(n/2)).

G.f.: 1/8*(2/q^2 -2 + 1/p - 1/q + 2*sqrt(p^2-2*x)/sqrt(q^2+2*x) - sqrt(2 + 2*p*q)/(p*q)), where p=sqrt(1+4*x) and q=sqrt(1-4*x). - Benedict W. J. Irwin, Aug 13 2016

Extensions

More terms from Robert G. Wilson v, Mar 24 2005