OFFSET
2,1
LINKS
Gheorghe Coserea, Table of n, a(n) for n = 2..307
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.
FORMULA
G.f.: (1 + 2*x)*F, where F = (1 - z + 2*z^3*x*(1 - x))/(2*z*(1 - z^2*x)) and z = 1 + 2*x + 6*x^2 + 34*x^3 + 254*x^4 + 2052*x^5 + 17332*x^6 + 151658*x^7 + ... satisfies 0 = x*(4*x^2 + 1)*z^4 - 4*x*(2*x - 1)*z^3 - 5*x*z^2 + 2*(2*x - 1)*z + 2. (see Theorem C in link)
G.f. y=A(x) satisfies:
0 = 8*y^4 + 4*(2*x + 1)*(2*x + 3)*y^3 - (x - 6)*(2*x + 1)^2*y^2 + (2*x + 1)^3*(18*x^2 - 16*x + 1)*y + x^2*(2*x + 1)^4*(27*x - 2).
0 = x^3*(2*x + 1)^4*(4*x^2 - 2*x + 1)*(108*x^2 - 304*x + 27)*(128*x^6 - 1504*x^5 + 5864*x^4 - 8282*x^3 + 4381*x^2 - 659*x + 60)*y'''' - x^2*(2*x + 1)^3*(417792*x^10 - 1973504*x^9 - 7891840*x^8 + 53958576*x^7 - 106786208*x^6 + 92663096*x^5 - 38721768*x^4 + 9604075*x^3 - 1447438*x^2 + 141966*x - 4860)*y''' + 3*x*(2*x + 1)^2*(163840*x^12 - 1929216*x^11 + 11348480*x^10 - 47888896*x^9 + 125855008*x^8 - 184580160*x^7 + 158611640*x^6 - 81013580*x^5 + 22892592*x^4 - 3821021*x^3 + 403960*x^2 - 23876*x + 120)*y'' - 12*(2*x + 1)*(163840*x^13 - 1888256*x^12 + 11294208*x^11 - 48430080*x^10 + 125093344*x^9 - 184709184*x^8 + 159190952*x^7 - 80413964*x^6 + 23140740*x^5 - 3792653*x^4 + 391233*x^3 - 28410*x^2 - 199*x + 30)*y' + 24*(163840*x^13 - 1847296*x^12 + 11198976*x^11 - 48855552*x^10 + 124699296*x^9 - 184627968*x^8 + 159583928*x^7 - 80114156*x^6 + 23238984*x^5 - 3787577*x^4 + 385076*x^3 - 30072*x^2 - 292*x + 40)*y.
a(n) ~ c / (sqrt(Pi) * n^(5/2) * r^n), where r = (76 - 7*sqrt(103))/54 and c = sqrt(3278181/(3125*(109592 + 10823*sqrt(103)))). - Vaclav Kotesovec, Aug 25 2018
EXAMPLE
A(x) = 2*x^2 + 9*x^3 + 30*x^4 + 154*x^5 + 986*x^6 + 6977*x^7 + 52590*x^8 + ...
PROG
(PARI)
F = (1 - z + 2*z^3*x*(1 - x))/(2*z*(1 - z^2*x));
G = x*(4*x^2 + 1)*z^4 - 4*x*(2*x - 1)*z^3 - 5*x*z^2 + 2*(2*x - 1)*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((1 + 2*x)*subst(F, 'z, Z(N+2)));
seq(23)
\\ test: y=Ser(seq(303))*x^2; 0 == 8*y^4 + 4*(2*x + 1)*(2*x + 3)*y^3 - (x - 6)*(2*x + 1)^2*y^2 + (2*x + 1)^3*(18*x^2 - 16*x + 1)*y + x^2*(2*x + 1)^4*(27*x - 2)
CROSSREFS
KEYWORD
nonn
AUTHOR
Gheorghe Coserea, Aug 16 2018
STATUS
approved