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

2, 12, 92, 800, 7554, 75664, 792448, 8595120, 95895816, 1095130728, 12753454896, 151017596448, 1814135701956, 22067487234504, 271407264938656, 3370796862212944, 42230992336570032, 533252038221313888, 6781213722509638192, 86790636905453265216

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

References

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

Links

• _Peter J. Taylor_, Table of n, a(n) for n = 0..500
• P. N. Rathie, The enumeration of Hamiltonian polygons in rooted planar triangulations, Discrete Math., 6 (1973), 163-168.

Programs

PARI

P(n,k) = k*(2*n+2*k-4)!*(2*n+k-1)!/((n+k-1)!*(n+k-2)!*n!*(n+k)!);
F(K, N=23) = {
  my(x='x + O('x^(K+1)), t='t + O('t^(N+1)),
     r='t*Ser(vector(N, n, sqr(binomial(2*n,n)/(n+1))),'t),
     p=x^3*Ser(apply(k->Ser(vector(N, n, P(n-1,k)),'t), [3..K])),
     s=serreverse(t*(1+r)), f=subst(subst(p, 't, s), 'x, 'x*s/'t));
  Vec(polcoeff(f,K));
};
F(4) \\ Gheorghe Coserea, Aug 18 2017

Formula

a(n) = f(n, 4) where f(n, k) is defined in A003122. - Sean A. Irvine, Feb 02 2015

Extensions

More terms and title clarified by Sean A. Irvine, Feb 02 2015