OFFSET
1,3
REFERENCES
Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, p. 419.
LINKS
Gheorghe Coserea, Table of n, a(n) for n = 1..126
M. Bodirsky, C. Groepl and M. Kang, Generating Labeled Planar Graphs Uniformly At Random, ICALP03 Eindhoven, LNCS 2719, Springer Verlag (2003), 1095 - 1107.
M. Bodirsky, C. Groepl and M. Kang, Generating Labeled Planar Graphs Uniformly At Random, Theoretical Computer Science, Volume 379, Issue 3, 15 June 2007, Pages 377-386.
O. Gimenez and M. Noy, Asymptotic enumeration and limit laws of planar graphs, arXiv:math/0501269 [math.CO], 2005.
FORMULA
This is generated by log(1+g(x)), where g(x) is the e.g.f. for labeled planar graphs, which may be computed from recurrences in Bodirsky et al. - Keith Briggs, Feb 04 2005
a(n) ~ c * n^(-7/2) * gamma^n * n!, where c = 0.00000410436110025...(A266392) and gamma = 27.2268777685...(A266390) (see Gimenez and Noy). - Gheorghe Coserea, Feb 24 2016
EXAMPLE
There are 4 connected labeled planar graphs on 3 nodes:
1-2-3,
1-3-2,
2-1-3 and
1-2
|/
3
PROG
(PARI)
Q(n, k) = { \\ c-nets with n-edges, k-vertices
if (k < 2+(n+2)\3 || k > 2*n\3, return(0));
sum(i=2, k, sum(j=k, n, (-1)^((i+j+1-k)%2)*binomial(i+j-k, i)*i*(i-1)/2*
(binomial(2*n-2*k+2, k-i)*binomial(2*k-2, n-j) -
4*binomial(2*n-2*k+1, k-i-1)*binomial(2*k-3, n-j-1))));
};
A100960_ser(N) = {
my(x='x+O('x^(3*N+1)), t='t+O('t^(N+4)),
q=t*x*Ser(vector(3*N+1, n, Polrev(vector(min(N+3, 2*n\3), k, Q(n, k)), 't))),
d=serreverse((1+x)/exp(q/(2*t^2*x) + t*x^2/(1+t*x))-1),
g2=intformal(t^2/2*((1+d)/(1+x)-1)));
serlaplace(Ser(vector(N, n, subst(polcoeff(g2, n, 't), 'x, 't)))*'x);
};
A096332_seq(N) = {
my(x='x+O('x^(N+3)), b=x^2/2+serconvol(Ser(A096331_seq(N))*x^3, exp(x)));
Vec(serlaplace(intformal(serreverse(x/exp(b'))/x)));
};
A096332_seq(15) \\ Gheorghe Coserea, Aug 10 2017
CROSSREFS
KEYWORD
nonn,hard
AUTHOR
Steven Finch, Aug 02 2004
EXTENSIONS
More terms from Keith Briggs, Feb 04 2005
More terms from Alois P. Heinz, Dec 30 2015
STATUS
approved