A255638 Number of n-node rooted trees with a forbidden limb of length 8.

1, 1, 2, 4, 9, 20, 48, 115, 285, 717, 1837, 4753, 12451, 32878, 87549, 234654, 632814, 1715446, 4672544, 12780511, 35091842, 96684570, 267223650, 740691451, 2058470490, 5734620428, 16011730686, 44799543203, 125587727914, 352696990890, 992169406959

Offset: 1

Author

Alois P. Heinz, Feb 28 2015

Keywords

• nonn

Comments

Any rootward 8-node path starting at a leaf contains the root or a branching node.

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..1000

Crossrefs

Column k=8 of A255636.

Programs

Maple

with(numtheory):
g:= proc(n, k) option remember; `if`(n=0, 1, add(add(d*(g(d-1, k)-
      `if`(d=k, 1, 0)), d=divisors(j))*g(n-j, k), j=1..n)/n)
    end:
a:= n-> g(n-1, 8):
seq(a(n), n=1..40);

Formula

a(n) ~ c * d^n / n^(3/2), where d = 2.95534758805897943282870965508636011438518881375... and c = 0.43969195292137654501555629074396535899093... . - Vaclav Kotesovec, Feb 28 2016