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A096331 Number of 2-connected planar graphs on n labeled nodes. 9
1, 10, 237, 10707, 774924, 78702536, 10273189176, 1631331753120, 304206135619160, 65030138045062272, 15659855107404275280, 4191800375194003211360, 1234179902360142341550240, 396280329098426228719121280, 137779269467538258010671193472 (list; graph; refs; listen; history; text; internal format)
OFFSET
3,2
COMMENTS
Recurrence known, see Bodirsky et al.
REFERENCES
Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, p. 419.
LINKS
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
Cf. A066537. Row sums of A100960.
Sequence in context: A188679 A167867 A268080 * A159497 A177595 A013922
KEYWORD
nonn
AUTHOR
Steven Finch, Aug 02 2004
EXTENSIONS
More terms from Gheorghe Coserea, Aug 05 2017
STATUS
approved

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Last modified June 15 19:43 EDT 2024. Contains 373410 sequences. (Running on oeis4.)