A052324 Number of increasing rooted trees with a forbidden limb of length 3.

0, 1, 1, 2, 5, 19, 90, 520, 3475, 26550, 228050, 2177020, 22860090, 261870070, 3249793360, 43432062300, 621911561150, 9498946124800, 154152712434600, 2648808048264400, 48043086765929200, 917249983543337400

Offset: 0

Author

Christian G. Bower, Dec 15 1999

Keywords

• nonn
• eigen

Comments

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

A rooted tree with a forbidden limb of length k is a rooted tree where the path from any leaf inward hits a branching node or the root within k steps.

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..240
• N. J. A. Sloane, Transforms
• Index entries for sequences related to rooted trees

Crossrefs

Cf. A002955, A002988-A002992, A052319-A052329.

Programs

Mathematica

CoefficientList[Assuming[{Element[x, Reals], x > 0}, Series[-Log[1-6^(1/3)*Gamma[1/3]/3 + 1/3*x*ExpIntegralE[2/3, x^3/6]], {x, 0, 20}]], x]*Range[0, 20]! (* Vaclav Kotesovec, Mar 28 2014 *)

PARI

{a(n)=local(A=x); for(i=0, n, A=intformal(exp(A-x^3/6+O(x^n)) )); n!*polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", ")) \\ Vaclav Kotesovec, Mar 28 2014

Formula

E.g.f. satisfies A'(x) = exp(A(x) - x^3/6). - corrected by Vaclav Kotesovec, Mar 28 2014

a(n) ~ d^n * (n-1)!, where d = 0.9546118344740519430556804... - Vaclav Kotesovec, Mar 28 2014

In closed form, d = 1/r, where r = 1.04754620033697244977759528695194261... is the root of the equation 1 = Integral_{x=0..r} exp(-x^3/6) dx. - Vaclav Kotesovec, Aug 21 2014