A003123 Number of Hamiltonian rooted triangulations with n internal nodes and 4 external nodes. (Formerly M2038)

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

Offset

0,1

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.

Formula

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

Prog

(C#) See A003122
(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

Crossrefs

Sequence in context: A333473 A103882 A304979 * A316143 A372202 A359692

Adjacent sequences: A003120 A003121 A003122 * A003124 A003125 A003126

Keyword

nonn

Author

N. J. A. Sloane

Extensions

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

Status

approved