A291842 a(n) is the number of labeled connected planar graphs with n edges.

1, 3, 17, 140, 1524, 20673, 336259, 6382302, 138525770, 3384987698, 91976075664, 2751117418712, 89832957177685, 3179833729806525, 121286809954760876, 4959277317653328656, 216402696660205555698, 10037527922988058277877, 493159461152794975438450, 25585023231409205439510792

Offset: 1

Author

Gheorghe Coserea, Sep 10 2017

Keywords

• nonn

Links

• Gheorghe Coserea, Table of n, a(n) for n = 1..123
• E. A. Bender, Z. Gao and N. C. Wormald, The number of labeled 2-connected planar graphs, Electron. J. Combin., 9 (2002), #R43.
• P. S. Kolesnikov and B. K. Sartayev, On the special identities of Gelfand--Dorfman algebras, arXiv:2105.13815 [math.RA], 2021.

Crossrefs

Cf. A096332, A290326, A291841.

Column sums of A288265.

Programs

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))));
};
seq(N) = {
my(x='x+O('x^(N+3)), t='t,
   q=t*x*Ser(vector(N, n, Polrev(vector(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)),
   b=t*'x^2/2 + 'x*Ser(vector(N+1, n, subst(polcoeff(g2, n, 't),'x,'t))),
   g1=intformal(serreverse('x/exp(b'))/'x),
   e1='x*Ser(vector(N, n, subst(polcoeff(serlaplace(g1), n, 't), 'x, 't))));
   Vec(subst(e1,'t,1));
};
seq(20)