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%I #11 Feb 28 2016 10:46:18
%S 1,1,2,4,9,20,48,115,285,717,1837,4753,12451,32878,87549,234654,
%T 632814,1715446,4672544,12780511,35091842,96684570,267223650,
%U 740691451,2058470490,5734620428,16011730686,44799543203,125587727914,352696990890,992169406959
%N Number of n-node rooted trees with a forbidden limb of length 8.
%C Any rootward 8-node path starting at a leaf contains the root or a branching node.
%H Alois P. Heinz, <a href="/A255638/b255638.txt">Table of n, a(n) for n = 1..1000</a>
%F a(n) ~ c * d^n / n^(3/2), where d = 2.95534758805897943282870965508636011438518881375... and c = 0.43969195292137654501555629074396535899093... . - _Vaclav Kotesovec_, Feb 28 2016
%p with(numtheory):
%p g:= proc(n, k) option remember; `if`(n=0, 1, add(add(d*(g(d-1, k)-
%p `if`(d=k, 1, 0)), d=divisors(j))*g(n-j, k), j=1..n)/n)
%p end:
%p a:= n-> g(n-1, 8):
%p seq(a(n), n=1..40);
%Y Column k=8 of A255636.
%K nonn
%O 1,3
%A _Alois P. Heinz_, Feb 28 2015
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