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

0,4

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

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

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 *)

Prog

(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

Crossrefs

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

Sequence in context: A323389 A268605 A205804 * A020115 A103816 A052169

Adjacent sequences: A052321 A052322 A052323 * A052325 A052326 A052327

Keyword

nonn,eigen

Author

Christian G. Bower, Dec 15 1999

Status

approved