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%I #6 Oct 06 2013 13:59:10
%S 1,1,2,4,10,24,62,160,425,1140,3105,8528,23643,66008,185526,524384,
%T 1489810,4251852,12184745,35048405,101156752,292865417,850314803,
%U 2475327088,7223400899,21126670372,61920289652,181838859665
%N Number of 1-2-3-4 trees with n edges and with thinning limbs. A 1-2-3-4 tree is an ordered tree with vertices of outdegree at most 4. A rooted tree with thinning limbs is such that if a node has k children, all its children have at most k children.
%C The sequences corresponding to k=2 (A090344), k=3 (A124497), k=4 (this A124499), k=5 (A124500), etc. approach sequence A124344, corresponding to ordered trees with thinning limbs.
%F In general, 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), M[1](z)=1/(1-z).
%o (PARI) {a(n)=local(k=4,M=1+x*O(x^n)); for(i=1,k,M=M*sum(j=0,n,binomial(i*j,j)/((i-1)*j+1)*(x^i*M^(i-1))^j)); polcoeff(M,n)} \\ _Paul D. Hanna_
%Y Cf. A090344, A124497, A124500, A124501, A124344.
%K nonn
%O 0,3
%A _Emeric Deutsch_ and _Louis Shapiro_, Nov 06 2006
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