A124501 Number of 1-2-3-4-5-6 trees with n edges and with thinning limbs. A 1-2-3-4-5-6 tree is an ordered tree with vertices of outdegree at most 6. 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, 25, 68, 186, 522, 1479, 4246, 12289, 35872, 105411, 311662, 926270, 2765778, 8292296, 24953437, 75338686, 228140842, 692733127, 2108652750, 6433255041, 19668210742, 60247367313, 184879648441, 568281131800

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 (A124499), k=5 (A124500), k=6 (this A124501), etc. approach sequence A124344, corresponding to ordered trees with thinning limbs.

Crossrefs

Cf. A090344, A124497, A124499, A124500, A124344.

Programs

PARI

{a(n)=local(k=6,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).