A071213 - OEIS

A071213

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

%I #7 Mar 25 2016 06:57:22

%S 1,4,34,436,7428,157368,3980688,116949600,3911421600,146673662400,

%T 6093249563520,277729608280320,13778539159795200,739059210587980800,

%U 42615627311477606400,2628646012982829772800,172704619437756321484800

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

%H C. Chauve, S. Dulucq and O. Guibert, <a href="http://www.cecm.sfu.ca/~cchauve/Publications/RR-1226-99.ps">Enumeration of some labeled trees</a>, research report RR-1226-99, LaBRI, Bordeaux I University, 1999.

%F 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

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

%o (Maxima)

%o C(x):=(1-sqrt(1-4*x))/(2);

%o F(x):=1+log(1-x)/x-log(1-x);

%o 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 */

%Y Cf. A000108, A000312, A033184.

%K easy,nonn

%O 2,2

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