A099553 Number of rooted 2-connected loopless 4-regular planar maps with n inner faces.

1, 2, 10, 42, 209, 1066, 5726, 31688, 180234, 1047356, 6198500, 37253790, 226891665, 1397880330, 8699804598, 54629525808, 345778883678, 2204263514460, 14142192816908, 91263177339092, 592069697914170, 3859674384409668, 25272938482712044

Offset

3,2

Comments

a(n) is also the number of rooted loopless planar maps with n-1 edges and no isthmuses. - Andrew Howroyd, Apr 01 2021

Links

Gheorghe Coserea, Table of n, a(n) for n = 3..305

Han Ren, Yanpei Liu and Zhaoxiang Li, Enumeration of 2-connected Loopless 4-regular Maps on the Plane, European J. Combin., 23 (2002), 93-111.

T. R. S. Walsh and A. B. Lehman, Counting rooted maps by genus. III: Nonseparable maps, J. Combinatorial Theory Ser. B 18 (1975), 222-259, eq (7).

Formula

From Gheorghe Coserea, Jul 12 2018: (Start)

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.)

G.f. y=A(x) satisfies:

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).

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.

(End)

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

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

Example

A(x) = x^3 + 2*x^4 + 10*x^5 + 42*x^6 + 209*x^7 + 1066*x^8 + 5726*x^9 + ...

Maple

A099553 := proc(n)
    local e;
    e := n-1 ;
    add(binomial(2*e-r, e-2-2*r)*2^r*binomial(2*e, r), r=0..floor(e/2-1)) ;
    %-3*add(binomial(2*e-r, e-3-2*r)*2^r*binomial(2*e, r), r=0..floor((e-3)/2)) ;
    %*2/e ;
end proc:
seq(A099553(n), n=3..30) ; # R. J. Mathar, Aug 28 2018

Mathematica

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];
Table[a[n], {n, 3, 30}] (* Jean-François Alcover, Feb 14 2023, after R. J. Mathar *)

Prog

(PARI)
F = (2*z^3*x^2 + (2*z^3 - 2*z)*x + (-z + 1))/(-2*z^3*x + 2*z);
G = x*(4*x + 1)*z^4 + 4*x*z^3 - 5*x*z^2 - 2*z + 2;
Z(N) = {
  my(z0=1+O('x^N), z1=0, n=1);
  while (n++,
    z1 = z0 - subst(G, 'z, z0)/subst(deriv(G, 'z), 'z, z0);
    if (z1 == z0, break()); z0 = z1);
  z0;
};
seq(N) = Vec(subst(F, 'z, Z(N+3)));
seq(23)
\\ 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)
\\ Gheorghe Coserea, Jul 13 2018
(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

Crossrefs

Row sums of A342980.

Cf. A000139, A058860, A290326.

Sequence in context: A177238 A084480 A309182 * A119694 A286760 A197048

Adjacent sequences: A099550 A099551 A099552 * A099554 A099555 A099556

Keyword

nonn

Author

N. J. A. Sloane, Nov 18 2004

Status

approved