A071213 Number of labeled planar trees with n nodes such that the root is smaller than all its children.

1, 4, 34, 436, 7428, 157368, 3980688, 116949600, 3911421600, 146673662400, 6093249563520, 277729608280320, 13778539159795200, 739059210587980800, 42615627311477606400, 2628646012982829772800, 172704619437756321484800

Offset: 2

Author

Cedric Chauve (chauve(AT)lacim.uqam.ca), May 16 2002

Keywords

• easy
• nonn

Links

• C. Chauve, S. Dulucq and O. Guibert, Enumeration of some labeled trees, research report RR-1226-99, LaBRI, Bordeaux I University, 1999.

Crossrefs

Cf. A000108, A000312, A033184.

Programs

Maple

n -> 2 * sum((n-1) * (n-2+k)! / (k! * (n-1-k)), k=0 .. n-2) - ((2*n-3)! / (n-2)!);

Maxima

C(x):=(1-sqrt(1-4*x))/(2);
F(x):=1+log(1-x)/x-log(1-x);
makelist(n!*coeff(taylor(F(C(x))+x*diff(F(C(x)),x),x,0,15),x,n),n,1,15); /*Vladimir Kruchinin, Mar 23 2016 */

Formula

E.g.f.: F(C(x))+x*[F(C(x))]', where C(x)/x is g.f. of Catalan numbers (A000108), F(x)=1+log(1-x)/x-log(1-x). - Vladimir Kruchinin, Mar 23 2016