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

1,2

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.

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

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 *)

Crossrefs

Cf. A060404, A033504.

Sequence in context: A002001 A113956 A323261 * A283679 A165328 A351282

Adjacent sequences: A103940 A103941 A103942 * A103944 A103945 A103946

Keyword

easy,nonn

Author

Valery A. Liskovets, Mar 17 2005

Extensions

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

Status

approved