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A096331 Number of 2-connected planar graphs on n labeled nodes. 8
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

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

Cf. A066537. Row sums of A100960.

Sequence in context: A188679 A167867 A268080 * A159497 A177595 A013922

Adjacent sequences:  A096328 A096329 A096330 * A096332 A096333 A096334

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 August 8 02:22 EDT 2020. Contains 336290 sequences. (Running on oeis4.)