|
%I #19 Aug 26 2016 12:22:54
%S 1,3,12,48,196,798,3248,13184,53416,216018,872344,3518496,14177528,
%T 57080572,229657792,923474944,3711572176,14911097514,59883185096,
%U 240416320928,964947251544,3872021946532,15533828715232,62306843932928
%N Number of unrooted two-vertex n-edge maps in the plane (planar with a distinguished outside face).
%D V. A. Liskovets and T. R. Walsh, Enumeration of unrooted maps on the plane, Rapport technique, UQAM, No. 2005-01, Montreal, Canada, 2005.
%H G. C. Greubel, <a href="/A103943/b103943.txt">Table of n, a(n) for n = 1..1000</a>
%H V. A. Liskovets and T. R. Walsh, <a href="http://dx.doi.org/10.1016/j.aam.2005.03.006">Counting unrooted maps on the plane</a>, Advances in Applied Math., 36, No.4 (2006), 364-387.
%F 2a(n) = 2^(2n-1) - binomial(2n-1, n-1) + binomial(n-1, floor(n/2)).
%F 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
%t 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 *)
%t 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 *)
%Y Cf. A060404, A033504.
%K easy,nonn
%O 1,2
%A _Valery A. Liskovets_, Mar 17 2005
%E More terms from _Robert G. Wilson v_, Mar 24 2005
|
