A003123 - OEIS

%I M2038 #31 May 30 2026 16:39:44

%S 2,12,92,800,7554,75664,792448,8595120,95895816,1095130728,

%T 12753454896,151017596448,1814135701956,22067487234504,

%U 271407264938656,3370796862212944,42230992336570032,533252038221313888,6781213722509638192,86790636905453265216

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

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Peter J. Taylor, <a href="/A003123/b003123.txt">Table of n, a(n) for n = 0..500</a>

%H P. N. Rathie, <a href="https://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 a(n) = f(n, 4) where f(n, k) is defined in A003122. - _Sean A. Irvine_, Feb 02 2015

%o (C#) See A003122

%o (PARI)

%o P(n,k) = k*(2*n+2*k-4)!*(2*n+k-1)!/((n+k-1)!*(n+k-2)!*n!*(n+k)!);

%o F(K, N=23) = {

%o   my(x='x + O('x^(K+1)), t='t + O('t^(N+1)),

%o      r='t*Ser(vector(N, n, sqr(binomial(2*n,n)/(n+1))),'t),

%o      p=x^3*Ser(apply(k->Ser(vector(N, n, P(n-1,k)),'t), [3..K])),

%o      s=serreverse(t*(1+r)), f=subst(subst(p, 't, s), 'x, 'x*s/'t));

%o   Vec(polcoeff(f,K));

%o };

%o F(4) \\ _Gheorghe Coserea_, Aug 18 2017

%Y Column k=1 of A391153.

%K nonn

%O 0,1

%A _N. J. A. Sloane_

%E More terms and title clarified by _Sean A. Irvine_, Feb 02 2015