OFFSET
3,2
COMMENTS
Recurrence known, see Bodirsky et al.
REFERENCES
Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, p. 419.
LINKS
Gheorghe Coserea, Table of n, a(n) for n = 3..126
E. A. Bender, Z. Gao and N. C. Wormald, The number of labeled 2-connected planar graphs, Electron. J. Combin., 9 (2002), #R43.
M. Bodirsky, C. Groepl and M. Kang, Generating Labeled Planar Graphs Uniformly At Random, ICALP03 Eindhoven, LNCS 2719, Springer Verlag (2003), 1095 - 1107.
O. Gimenez and M. Noy, Asymptotic enumeration and limit laws of planar graphs, arXiv:math/0501269 [math.CO], 2005.
FORMULA
a(n) ~ g * n^(-7/2) * r^n * n!, where g=0.00000370445941594... (A291835) and r=26.1841125556... (A291836) (see Bender link). - Gheorghe Coserea, Sep 03 2017
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);
};
Vec(subst(A100960_ser(20), 't, 1)) \\ Gheorghe Coserea, Aug 10 2017
CROSSREFS
KEYWORD
nonn
AUTHOR
Steven Finch, Aug 02 2004
EXTENSIONS
More terms from Gheorghe Coserea, Aug 05 2017
STATUS
approved