%I A124497
%S A124497 1,1,2,4,9,20,48,116,288,724,1849,4768,12423,32628,86342,229952,616042,
%T A124497 1659012,4489101,12199521,33284546,91140797,250396629,690043032,
%U A124497 1907022197,5284167884,14677681554,40862469713,114001697975
%N A124497 Number of 1-2-3 trees with n edges and with thinning limbs. A 1-2-3 tree 
               is an ordered tree with vertices of outdegree at most 3. A rooted 
               tree with thinning limbs is such that if a node has k children, all 
               its children have at most k children.
%F A124497 G.f.=H*T(H^2*z^3), where T=2/sqrt(3*x)*sin((1/3)*arcsin(sqrt(27*x/4))) 
               (solution of T=1+zT^3, T(0)=1), H=C(z^2/(1-z))/(1-z) and C(x)=[1-sqrt(1-4x)]/
               (2x) is the Catalan function. More generally, if M[k](z) is the g.f. 
               of the 1-2-...-k trees with thinning limbs and C[k](z)=1+z*{C[k](z)}^k 
               is the g.f. of the k-ary trees, then M[k](z)=M[k-1](z)C[k](M[k-1]^(k-1)*z^k).
%p A124497 C:=x->(1-sqrt(1-4*x))/2/x: T:=x->2/sqrt(3*x)*sin((1/3)*arcsin(sqrt(27*x/
               4))): M2:=C(z^2/(1-z))/(1-z): G:=M2*T(M2^2*z^3): Gser:=series(G,z=0,
               40): seq(coeff(Gser,z,n),n=0..33);
%Y A124497 Cf. A090344, A124344.
%Y A124497 Sequence in context: A145549 A145550 A000081 this_sequence A093637 A068051 
               A032289
%Y A124497 Adjacent sequences: A124494 A124495 A124496 this_sequence A124498 A124499 
               A124500
%K A124497 nonn
%O A124497 0,3
%A A124497 Emeric Deutsch and Louis Shapiro (deutsch(AT)duke.poly.edu, lshapiro(AT)Howard.edu), 
               Nov 04 2006