A131638 Increasing binary trees having exactly two vertices with outdegree 1.

1, 11, 180, 4288, 141584, 6213288, 350400832, 24718075136, 2133652515072, 221311262045440, 27166907582280704, 3895974311462313984, 645512064907811491840, 122381396964887716078592, 26325690425815766552887296, 6377608610246241663568248832

Offset: 1

Author

Wenjin Woan, Oct 03 2007

Keywords

• nonn

Links

• M. P. Develin and S. P. Sullivant, Markov Bases of Binary Graph Models, Annals of Combinatorics 7 (2003) 441-466.
• Christiane Poupard, Deux propriétés des arbres binaires ordonnés stricts, Europ. J. Combin., vol. 10, 1989, pp. 369-374.

Crossrefs

Cf. A002105, A094503.

Programs

Mathematica

Table[n!*SeriesCoefficient[1/2*(-((x*Sec[x/Sqrt[2]]^2 *Tan[x/Sqrt[2]]) /Sqrt[2]) + 3*Sec[x/Sqrt[2]]^2 *Tan[x/Sqrt[2]]^2), {x, 0, n}], {n, 2, 40, 2}] (* Vaclav Kotesovec after Michel Marcus, Sep 25 2013 *)

PARI

lista(m) = { default(realprecision, 30); x = y + O(y^m); egf = (3*tan(x/sqrt(2))^2/cos(x/sqrt(2))^2-x*tan(x/sqrt(2))/(sqrt(2)*cos(x/sqrt(2))^2))/2; forstep (n=2, m, 2, print1(round(n!*polcoeff(egf, n, y)), ", "));}  \\ Michel Marcus, Mar 03 2013

Formula

E.g.f.: (3*sec(x/sqrt(2))^2*tan(x/sqrt(2))^2-x*sec(x/sqrt(2))^2*tan(x/sqrt(2))/(sqrt(2)))/2. - Michel Marcus, Mar 03 2013

a(n) ~ (2*n)! * 2^(n+6)*n^3/Pi^(2*n+4). - Vaclav Kotesovec, Sep 25 2013

From Klaus K Haverkamp, Jul 02 2023: (Start)

a(n) = (A002105(n+2) - (n+1)*A002105(n+1))/2.

a(n) = A094503(2n+1,n). (End)

Extensions

More terms from Michel Marcus, Mar 03 2013