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

%I A052325 #18 Mar 01 2016 04:46:00
%S A052325 1,1,1,1,1,2,4,8,15,30,60,122,249,513,1061,2210,4620,9708,20472,43337,
%T A052325 92023,196018,418653,896485,1924154,4139014,8921349,19266067,41679483,
%U A052325 90318082,196020800,426055601,927317334,2020949226,4409764169
%N A052325 Number of asymmetric rooted trees with a forbidden limb of length 3.
%C A052325 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.
%H A052325 Alois P. Heinz, <a href="/A052325/b052325.txt">Table of n, a(n) for n = 1..1000</a>
%H A052325 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%H A052325 <a href="/index/Ro#rooted">Index entries for sequences related to rooted trees</a>
%F A052325 a(n) satisfies a = SHIFT_RIGHT(WEIGH(a-b)) where b(3)=1, b(k)=0 if k != 3.
%F A052325 a(n) ~ c * d^n / n^(3/2), where d = 2.27671458388797627098091744865..., c = 0.2935911773459468433271794078... . - _Vaclav Kotesovec_, Aug 25 2014
%p A052325 b:= proc(n, i) option remember;
%p A052325       `if`(n=0, 1, `if`(i<1, 0, add(binomial(a(i)-
%p A052325       `if`(i=3, 1, 0), j)*b(n-i*j, i-1), j=0..n/i)))
%p A052325     end:
%p A052325 a:= n-> `if`(n<1, 1, b(n-1, n-1)):
%p A052325 seq(a(n), n=1..50);  # _Alois P. Heinz_, Jul 06 2014
%t A052325 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}]]];
%t A052325 a[n_] := If[n<1, 1, b[n-1, n-1]];
%t A052325 Table[a[n], {n, 1, 50}] (* _Jean-François Alcover_, Mar 01 2016, after _Alois P. Heinz_ *)
%Y A052325 Cf. A002955, A002988-A002992, A052318-A052329.
%K A052325 nonn,eigen
%O A052325 1,6
%A A052325 _Christian G. Bower_, Dec 15 1999