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A003123
Number of Hamiltonian rooted triangulations with n internal nodes and 4 external nodes.
3
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
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
Column k=1 of A391153.
Sequence in context: A333473 A103882 A304979 * A316143 A372202 A359692
KEYWORD
nonn
EXTENSIONS
More terms and title clarified by Sean A. Irvine, Feb 02 2015
STATUS
approved