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A096332 Number of connected planar graphs on n labeled nodes. 8
1, 1, 4, 38, 727, 26013, 1597690, 149248656, 18919743219, 3005354096360, 569226803220234, 124594074249852576, 30861014504270954737, 8520443838646833231236, 2592150684565935977152860, 861079753184429687852978432, 310008316267496041749182487881 (list; graph; refs; listen; history; text; internal format)
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);

};

A096331_seq(N) = Vec(subst(A100960_ser(N+2), 't, 1));

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

Cf. A066537, A096331, A266390, A266392, A267411.

Sequence in context: A201861 A171779 A171203 * A084284 A084285 A084286

Adjacent sequences:  A096329 A096330 A096331 * A096333 A096334 A096335

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

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Last modified May 9 12:58 EDT 2021. Contains 343742 sequences. (Running on oeis4.)