A108525 Number of increasing ordered rooted trees with n generators.

1, 3, 27, 429, 9609, 277107, 9772803, 407452221, 19604840481, 1069202914083, 65177482634667, 4391636680582029, 324102772814580729, 25999541378465556627, 2252597527900572815763, 209625760563134613131421

Offset: 1

Author

Christian G. Bower, Jun 07 2005

Keywords

• nonn

Comments

A generator is a leaf or a node with just one child.

In an increasing rooted tree, nodes are numbered and numbers increase as you move away from root.

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 1..200
• Index entries for sequences related to rooted trees

Crossrefs

Cf. A108521-A108529, A001147.

Programs

Mathematica

Rest[CoefficientList[InverseSeries[Series[(Log[1-x]+7*Log[1+x]+2/(x-1))/4+1/2,{x,0,20}],x],x]*Range[0,20]!] (* Vaclav Kotesovec, Feb 20 2014 *)

PARI

{a(n)=local(A=x); for(i=1, n, A=intformal((A-1)^2 * (1+A) /(1 - 4*A + 2*A^2)+O(x^n))); n!*polcoeff(A, n)};
for(n=1, 20, print1(a(n), ", ")); /* Vaclav Kotesovec, Feb 20 2014 */

Formula

E.g.f. satisfies: A(x) = -1 + 2*A'(x) - A'(x)/(1-A(x))^2, corrected by Vaclav Kotesovec and Paul D. Hanna, Feb 20 2014

A(x) = Series_Reversion( (log(1-x) + 7*log(1+x) + 2/(x-1))/4 + 1/2). - Vaclav Kotesovec, Feb 20 2014

a(n) ~ sqrt(4-sqrt(2)) * 2^(3*n-13/4) * n^(n-1) / (exp(n) * (4-4*sqrt(2)-log(2)+14*log(2-1/sqrt(2)))^(n-1/2)). - Vaclav Kotesovec, Feb 20 2014