A244457 Number of unlabeled rooted trees with n nodes such that the minimal outdegree of inner nodes equals 3.

1, 0, 0, 1, 2, 2, 4, 7, 12, 20, 34, 56, 98, 167, 284, 484, 835, 1433, 2467, 4250, 7345, 12700, 22004, 38154, 66266, 115224, 200623, 349654, 610126, 1065739, 1863547, 3261672, 5714277, 10020092, 17586014, 30890654, 54305289, 95542387, 168221056, 296401979

Offset: 4

Author

Joerg Arndt and Alois P. Heinz, Jun 29 2014

Keywords

• nonn

Examples

			a(7) = 1:
      o
     /|\
    o o o
   /|\
  o o o
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 4..1000

Crossrefs

Column k=3 of A244454.

Cf. A244532.

Programs

Maple

b:= proc(n, i, t, k) option remember; `if`(n=0, `if`(t in [0, k],
      1, 0), `if`(i<1 or t>n, 0, add(binomial(b((i-1)$2, k$2)+j-1, j)*
      b(n-i*j, i-1, max(0,t-j), k), j=0..n/i)))
    end:
a:= n-> b(n-1$2, 3$2) -b(n-1$2, 4$2):
seq(a(n), n=4..45);

Mathematica

b[n_, i_, t_, k_] := b[n, i, t, k] = If[n == 0, If[t == 0 || t == k, 1, 0], If[i < 1, 0, Sum[Binomial[b[i - 1, i - 1, k, k] + j - 1, j]*b[n - i*j, i - 1, Max[0, t - j], k], {j, 0, n/i}]] // FullSimplify]; a[n_] := b[n - 1, n - 1, 3, 3] - b[n - 1, n - 1, 4, 4]; Table[a[n], {n, 4, 45}] (* Jean-François Alcover, Feb 06 2015, after Maple *)

Formula

a(n) ~ c * d^n / n^(3/2), where d = 1.8239199077079..., c = 0.49573400799... . - Vaclav Kotesovec, Jul 11 2014