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A003123 Number of Hamiltonian rooted triangulations with n internal nodes and 4 external nodes.

%I A003123 M2038 #25 Aug 21 2017 22:35:39
%S A003123 2,12,92,800,7554,75664,792448,8595120,95895816,1095130728,
%T A003123 12753454896,151017596448,1814135701956,22067487234504,
%U A003123 271407264938656,3370796862212944,42230992336570032,533252038221313888,6781213722509638192,86790636905453265216
%N A003123 Number of Hamiltonian rooted triangulations with n internal nodes and 4 external nodes.
%D A003123 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A003123 _Peter J. Taylor_, <a href="/A003123/b003123.txt">Table of n, a(n) for n = 0..500</a>
%H A003123 P. N. Rathie, <a href="http://dx.doi.org/10.1016/0012-365X(73)90046-0">The enumeration of Hamiltonian polygons in rooted planar triangulations</a>, Discrete Math., 6 (1973), 163-168.
%F A003123 a(n) = f(n, 4) where f(n, k) is defined in A003122. - _Sean A. Irvine_, Feb 02 2015
%o A003123 (C#) See A003122
%o A003123 (PARI)
%o A003123 P(n,k) = k*(2*n+2*k-4)!*(2*n+k-1)!/((n+k-1)!*(n+k-2)!*n!*(n+k)!);
%o A003123 F(K, N=23) = {
%o A003123   my(x='x + O('x^(K+1)), t='t + O('t^(N+1)),
%o A003123      r='t*Ser(vector(N, n, sqr(binomial(2*n,n)/(n+1))),'t),
%o A003123      p=x^3*Ser(apply(k->Ser(vector(N, n, P(n-1,k)),'t), [3..K])),
%o A003123      s=serreverse(t*(1+r)), f=subst(subst(p, 't, s), 'x, 'x*s/'t));
%o A003123   Vec(polcoeff(f,K));
%o A003123 };
%o A003123 F(4) \\ _Gheorghe Coserea_, Aug 18 2017
%K A003123 nonn
%O A003123 0,1
%A A003123 _N. J. A. Sloane_
%E A003123 More terms and title clarified by _Sean A. Irvine_, Feb 02 2015