A052325 Number of asymmetric rooted trees with a forbidden limb of length 3.

1, 1, 1, 1, 1, 2, 4, 8, 15, 30, 60, 122, 249, 513, 1061, 2210, 4620, 9708, 20472, 43337, 92023, 196018, 418653, 896485, 1924154, 4139014, 8921349, 19266067, 41679483, 90318082, 196020800, 426055601, 927317334, 2020949226, 4409764169

Offset: 1

Author

Christian G. Bower, Dec 15 1999

Keywords

• nonn
• eigen

Comments

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

• Alois P. Heinz, Table of n, a(n) for n = 1..1000
• N. J. A. Sloane, Transforms
• Index entries for sequences related to rooted trees

Crossrefs

Cf. A002955, A002988-A002992, A052318-A052329.

Programs

Maple

b:= proc(n, i) option remember;
      `if`(n=0, 1, `if`(i<1, 0, add(binomial(a(i)-
      `if`(i=3, 1, 0), j)*b(n-i*j, i-1), j=0..n/i)))
    end:
a:= n-> `if`(n<1, 1, b(n-1, n-1)):
seq(a(n), n=1..50);  # Alois P. Heinz, Jul 06 2014

Mathematica

b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, Sum[Binomial[a[i]- If[i==3, 1, 0], j]*b[n-i*j, i-1], {j, 0, n/i}]]];
a[n_] := If[n<1, 1, b[n-1, n-1]];
Table[a[n], {n, 1, 50}] (* Jean-François Alcover, Mar 01 2016, after Alois P. Heinz *)

Formula

a(n) satisfies a = SHIFT_RIGHT(WEIGH(a-b)) where b(3)=1, b(k)=0 if k != 3.

a(n) ~ c * d^n / n^(3/2), where d = 2.27671458388797627098091744865..., c = 0.2935911773459468433271794078... . - Vaclav Kotesovec, Aug 25 2014