A124499 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.

1, 1, 2, 4, 10, 24, 62, 160, 425, 1140, 3105, 8528, 23643, 66008, 185526, 524384, 1489810, 4251852, 12184745, 35048405, 101156752, 292865417, 850314803, 2475327088, 7223400899, 21126670372, 61920289652, 181838859665

Offset: 0

Author

Emeric Deutsch and Louis Shapiro, Nov 06 2006

Keywords

• nonn

Comments

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.

Crossrefs

Cf. A090344, A124497, A124500, A124501, A124344.

Programs

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

Formula

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).