A124973 a(n) = Sum_{k=0..(n-2)/2} a(k)a*(n-1-k), with a(0) = a(1) = 1.

1, 1, 1, 1, 2, 3, 6, 11, 22, 42, 87, 174, 365, 745, 1587, 3303, 7103, 14974, 32477, 69284, 151172, 325077, 713400, 1545719, 3406989, 7423648, 16429555, 35992438, 79912474, 175785514, 391488688, 864591621, 1930333822, 4276537000

Offset: 0

Author

Franklin T. Adams-Watters, Nov 14 2006

Keywords

• easy
• nonn

Comments

Number of unordered rooted trees with all outdegrees <= 2 and, if a node has two subtrees, they have a different number of nodes (equivalently, ordered rooted trees where the left subtree has more nodes than the right subtree).

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A000108, A000992, A001190, A032305.

Programs

Maple

a:= proc(n) option remember;
      if n<2 then 1
    else add(a(j)*a(n-j-1), j=0..floor((n-2)/2))
      fi
    end:
seq(a(n), n=0..40); # G. C. Greubel, Nov 19 2019

Mathematica

a[n_]:= a[n]= If[n<2, 1, Sum[a[j]*a[n-j-1], {j, 0, (n-2)/2}]]; Table[a[n], {n, 0, 40}] (* G. C. Greubel, Nov 19 2019 *)

PARI

a(n) = if(n<2, 1, sum(j=0, (n-2)\2, a(j)*a(n-j-1))); \\ G. C. Greubel, Nov 19 2019

Sage

@CachedFunction
def a(n):
    if (n<2): return 1
    else: return sum(a(j)*a(n-j-1) for j in (0..floor((n-2)/2)))
[a(n) for n in (0..40)] # G. C. Greubel, Nov 19 2019

Formula

Lim_{n->infinity} a(n)^(1/n) = 2.327833478... - Vaclav Kotesovec, Nov 20 2019