A308524 - OEIS

%I #20 Sep 03 2019 18:40:04

%S 0,0,1,2,11,40,166,658,2647,10592,42446,169972,680670,2725320,

%T 10910992,43678882,174843151,699839680,2801078662,11210671612,

%U 44866276906,179552951440,718539964132,2875389341332,11506176209206,46042099714240,184234059839116,737184620655368

%N Number of essentially 3-connected rooted toroidal maps with n edges.

%H Michael De Vlieger, <a href="/A308524/b308524.txt">Table of n, a(n) for n = 0..1000</a>

%H Nicolas Bonichon, Éric Fusy, Benjamin Lévêque, <a href="https://arxiv.org/abs/1907.04016">A bijection for essentially 3-connected toroidal maps</a>, arXiv:1907.04016 [math.CO], 2019.

%F G.f.: A^2*(1+A)/((1+2*A)*(1-A)^2*(1+3*A)) where A=x*(1+A)^2.

%F G.f.: x*(1 + 8*x + (2*x - 1)*sqrt(1 - 4*x))/(2*(2 + x)*(1 - 4*x)*(3 + 4*x)). - _Vaclav Kotesovec_, Jun 25 2019

%F a(n) ~ 2^(2*n - 3) / 3. - _Vaclav Kotesovec_, Jun 25 2019

%p dev_A := 0; n := 20; dev_A := series(RootOf(A-x*(1+A)^2, A), x = 0, n+1);

%p seq(coeff(series(subs(A = dev_A, A^2*(1+A)/((1+2*A)*(1-A)^2*(1+3*A))), x, n+1), x, k), k = 0 .. n);

%p # second Maple program:

%p a:= proc(n) option remember; `if`(n<6, [0, 0, 1, 2, 11, 40][n+1],

%p      ((37*n^2-258*n+401)*a(n-1)-6*(2*n^2-25*n+88)*a(n-2)

%p       -48*(3*n^2-23*n+45)*a(n-3)-32*(n-4)*(2*n-7)*a(n-4))

%p       /((6*(n-1))*(n-5)))

%p     end:

%p seq(a(n), n=0..30);  # _Alois P. Heinz_, Jun 07 2019

%t CoefficientList[Series[x*(1 + 8*x + (2*x - 1)*Sqrt[1 - 4*x])/(2*(2 + x)*(1 - 4*x)*(3 + 4*x)), {x, 0, 30}], x] (* _Vaclav Kotesovec_, Jun 25 2019 *)

%Y Cf. A308523, A308526, A289208, A006422.

%K nonn

%O 0,4

%A _Nicolas Bonichon_, Jun 05 2019