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

%I A291842 #12 Sep 13 2021 20:59:00
%S A291842 1,3,17,140,1524,20673,336259,6382302,138525770,3384987698,
%T A291842 91976075664,2751117418712,89832957177685,3179833729806525,
%U A291842 121286809954760876,4959277317653328656,216402696660205555698,10037527922988058277877,493159461152794975438450,25585023231409205439510792
%N A291842 a(n) is the number of labeled connected planar graphs with n edges.
%H A291842 Gheorghe Coserea, <a href="/A291842/b291842.txt">Table of n, a(n) for n = 1..123</a>
%H A291842 E. A. Bender, Z. Gao and N. C. Wormald, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v9i1r43">The number of labeled 2-connected planar graphs</a>, Electron. J. Combin., 9 (2002), #R43.
%H A291842 P. S. Kolesnikov and B. K. Sartayev, <a href="https://arxiv.org/abs/2105.13815">On the special identities of Gelfand--Dorfman algebras</a>, arXiv:2105.13815 [math.RA], 2021.
%o A291842 (PARI)
%o A291842 Q(n,k) = { \\ c-nets with n-edges, k-vertices
%o A291842   if (k < 2+(n+2)\3 || k > 2*n\3, return(0));
%o A291842   sum(i=2, k, sum(j=k, n, (-1)^((i+j+1-k)%2)*binomial(i+j-k,i)*i*(i-1)/2*
%o A291842   (binomial(2*n-2*k+2,k-i)*binomial(2*k-2, n-j) -
%o A291842   4*binomial(2*n-2*k+1, k-i-1)*binomial(2*k-3, n-j-1))));
%o A291842 };
%o A291842 seq(N) = {
%o A291842 my(x='x+O('x^(N+3)), t='t,
%o A291842    q=t*x*Ser(vector(N, n, Polrev(vector(2*n\3, k, Q(n,k)),t))),
%o A291842    d=serreverse((1+x)/exp(q/(2*t^2*x) + t*x^2/(1+t*x))-1),
%o A291842    g2=intformal(t^2/2*((1+d)/(1+x)-1)),
%o A291842    b=t*'x^2/2 + 'x*Ser(vector(N+1, n, subst(polcoeff(g2, n, 't),'x,'t))),
%o A291842    g1=intformal(serreverse('x/exp(b'))/'x),
%o A291842    e1='x*Ser(vector(N, n, subst(polcoeff(serlaplace(g1), n, 't), 'x, 't))));
%o A291842    Vec(subst(e1,'t,1));
%o A291842 };
%o A291842 seq(20)
%Y A291842 Cf. A096332, A290326, A291841.
%Y A291842 Column sums of A288265.
%K A291842 nonn
%O A291842 1,2
%A A291842 _Gheorghe Coserea_, Sep 10 2017