editing

approved

(PARI) {a(n)=local(k=5, 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_)

proposed

editing

editing

proposed

(PARI) {a(n)=local(k=5, 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_)

_Emeric Deutsch _ and _Louis Shapiro (deutsch(AT)duke.poly.edu, lshapiro(AT)Howard.edu), _, Nov 06 2006

approved

editing

(PARI) {a(n)=local(k=5, 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)

nonn,new

nonn

Number of 1-2-3-4-5 trees with n edges and with thinning limbs. A 1-2-3-4-5 tree is an ordered tree with vertices of outdegree at most 5. 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, 67, 180, 495, 1375, 3871, 10993, 31493, 90843, 263686, 769466, 2256135, 6643082, 19634705, 58232350, 173242381, 516860717, 1546035258, 4635543843, 13929569399, 41943013047, 126532961332, 382396277940

0,3

The sequences corresponding to k=2 (A090344), k=3 (A124497), k=4 (A124499), k=5 (this A124500), etc. approach sequence A124344, corresponding to ordered trees with thinning limbs.

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).

(PARI) {a(n)=local(k=5, 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)

Cf. A090344, A124497, A124499, A124501, A124344.

nonn

Emeric Deutsch and Louis Shapiro (deutsch(AT)duke.poly.edu, lshapiro(AT)Howard.edu), Nov 06 2006

approved