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%I #51 Feb 14 2023 07:42:37
%S 1,2,10,42,209,1066,5726,31688,180234,1047356,6198500,37253790,
%T 226891665,1397880330,8699804598,54629525808,345778883678,
%U 2204263514460,14142192816908,91263177339092,592069697914170,3859674384409668,25272938482712044
%N Number of rooted 2-connected loopless 4-regular planar maps with n inner faces.
%C a(n) is also the number of rooted loopless planar maps with n-1 edges and no isthmuses. - _Andrew Howroyd_, Apr 01 2021
%H Gheorghe Coserea, <a href="/A099553/b099553.txt">Table of n, a(n) for n = 3..305</a>
%H Han Ren, Yanpei Liu and Zhaoxiang Li, <a href="https://doi.org/10.1006/eujc.2001.0533">Enumeration of 2-connected Loopless 4-regular Maps on the Plane</a>, European J. Combin., 23 (2002), 93-111.
%H T. R. S. Walsh and A. B. Lehman, <a href="http://dx.doi.org/10.1016/0095-8956(75)90050-7">Counting rooted maps by genus. III: Nonseparable maps</a>, J. Combinatorial Theory Ser. B 18 (1975), 222-259, eq (7).
%F From _Gheorghe Coserea_, Jul 12 2018: (Start)
%F G.f. A(x) = (2*z^3*x^2 + (2*z^3 - 2*z)*x + (-z + 1))/(-2*z^3*x + 2*z), where z = 1 + 2*x^2 + 6*x^3 + 34*x^4 + 176*x^5 + 1004*x^6 + ... satisfies 0 = x*(4*x + 1)*z^4 + 4*x*z^3 - 5*x*z^2 - 2*z + 2. (See Theorem D in reference.)
%F G.f. y=A(x) satisfies:
%F 0 = 8*y^4 + (32*x + 12)*y^3 + (48*x^2 + 23*x + 6)*y^2 + (32*x^3 + 10*x^2 - 10*x + 1)*y + x^3*(8*x - 1).
%F 0 = x^3*(2*x + 1)*(49*x - 18)*(196*x - 27)*y'''' + x^2*(96040*x^3 - 27587*x^2 - 9297*x + 972)*y''' + (72030*x^4 - 36309*x^3 + 2010*x^2 - 864*x)*y'' - 6*(8*x + 3)*(49*x + 12)*y' + (2352*x + 576)*y.
%F (End)
%F Conjecture: 3*n *(3*n-1) *(5*n-8) *(3*n-2)*a(n) -(n-2) *(2*n-3) *(355*n^2 -703*n +300)*a(n-1) -98*(n-2) *(5*n-3) *(2*n-3) *(2*n-5) *a(n-2)=0. - _R. J. Mathar_, Aug 28 2018
%F G.f.: x*(A(x) - 1) where A(x) satisfies A(x) = G(x*A(x)^2) and (G(x) + 2*x - 1)/x is the g.f. of A000139. - _Andrew Howroyd_, Apr 06 2021
%e A(x) = x^3 + 2*x^4 + 10*x^5 + 42*x^6 + 209*x^7 + 1066*x^8 + 5726*x^9 + ...
%p A099553 := proc(n)
%p local e;
%p e := n-1 ;
%p add(binomial(2*e-r,e-2-2*r)*2^r*binomial(2*e,r),r=0..floor(e/2-1)) ;
%p %-3*add(binomial(2*e-r,e-3-2*r)*2^r*binomial(2*e,r),r=0..floor((e-3)/2)) ;
%p %*2/e ;
%p end proc:
%p seq(A099553(n),n=3..30) ; # _R. J. Mathar_, Aug 28 2018
%t a[n_] := Module[{e, s}, e = n-1; s = Sum[Binomial[2e-r, e-2-2r]*2^r*Binomial[2e, r], {r, 0, Floor[e/2-1]}]; s = s-3*Sum[Binomial[2e-r, e-3-2r]*2^r*Binomial[2e, r], {r, 0, Floor[(e-3)/2]}]; s=2s/e];
%t Table[a[n], {n, 3, 30}] (* _Jean-François Alcover_, Feb 14 2023, after _R. J. Mathar_ *)
%o (PARI)
%o F = (2*z^3*x^2 + (2*z^3 - 2*z)*x + (-z + 1))/(-2*z^3*x + 2*z);
%o G = x*(4*x + 1)*z^4 + 4*x*z^3 - 5*x*z^2 - 2*z + 2;
%o Z(N) = {
%o my(z0=1+O('x^N), z1=0, n=1);
%o while (n++,
%o z1 = z0 - subst(G, 'z, z0)/subst(deriv(G,'z), 'z, z0);
%o if (z1 == z0, break()); z0 = z1);
%o z0;
%o };
%o seq(N) = Vec(subst(F, 'z, Z(N+3)));
%o seq(23)
%o \\ test: y = Ser(seq(303))*'x^3; 0 == 8*y^4 + (32*x + 12)*y^3 + (48*x^2 + 23*x + 6)*y^2 + (32*x^3 + 10*x^2 - 10*x + 1)*y + x^3*(8*x - 1)
%o \\ _Gheorghe Coserea_, Jul 13 2018
%o (PARI) seq(n)={my(g=1+x*sum(n=1,n,x^n*binomial(3*n, n)*2/((n+1)*(2*n+1))) + O(x*x^n)); Vec(-1 + sqrt(serreverse(x/g^2)/x))} \\ _Andrew Howroyd_, Apr 06 2021
%Y Row sums of A342980.
%Y Cf. A000139, A058860, A290326.
%K nonn
%O 3,2
%A _N. J. A. Sloane_, Nov 18 2004
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