%I A025225
%S A025225 2,4,16,80,448,2688,16896,109824,732160,4978688,34398208,240787456,
%T A025225 1704034304,12171673600,87636049920,635361361920,4634400522240,
%U A025225 33985603829760,250420238745600,1853109766717440
%N A025225 a(n) = a(1)*a(n-1) + a(2)*a(n-2) + ...+ a(n-1)*a(1) for n >= 2. Also 
               a(n) = (2^n)*C(n-1), where C = A000108 (Catalan numbers).
%C A025225 Number of generators of degree n of the Hopf algebra of 2-colored planar 
               binary trees. Also, dimensions of the graded components of the primitive 
               Lie algebra of the same Hopf algebra. - Jean-Yves Thibon (jyt(AT)univ-mlv.fr), 
               Jun 26 2008
%H A025225 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=653">
               Encyclopedia of Combinatorial Structures 653</a>
%H A025225 J.-C. Novelli and J.-Y. Thibon, <a href="http://fr.arxiv.org/abs/0806.3682">
               Free quasi-symmetric functions and descent algebras for wreath products 
               and noncommutative multi-symmetric functions</a>
%F A025225 G.f.: (1-sqrt(1-8*x))/2 - Michael Somos, Jun 08, 2000.
%F A025225 Given g.f. C(x) and given A(x)= g.f. of A100238, then B(x)=A(x)-1-x satisfies 
               B(x)=x-C(x*B(x)). - Michael Somos Sep 07 2005
%t A025225 InverseSeries[Series[y/2-y^2/2, {y, 0, 24}], x] (* then A(x)=y(x) *) 
               - Len Smiley Apr 13 2000
%o A025225 (PARI) a(n)=polcoeff((1-sqrt(1-8*x+x*O(x^n)))/2,n)
%Y A025225 Essentially identical to A115125.
%Y A025225 Sequence in context: A102736 A103619 A027436 this_sequence A115125 A000831 
               A000090
%Y A025225 Adjacent sequences: A025222 A025223 A025224 this_sequence A025226 A025227 
               A025228
%K A025225 nonn
%O A025225 1,1
%A A025225 Clark Kimberling (ck6(AT)evansville.edu)
%E A025225 Typo in definition corrected by R. J. Mathar, Aug 11 2008