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A108530 Number of rooted identity trees with n internal (non-leaf) nodes.

%I A108530 #20 Mar 03 2016 12:12:17
%S A108530 1,1,2,4,12,34,110,364,1248,4356,15520,56022,204726,755472,2812004,
%T A108530 10543718,39791070,151022006,576090250,2207493080,8493196536,
%U A108530 32797115398,127071214442,493831241234,1924504466246,7519182311366,29447430754182,115577336981932
%N A108530 Number of rooted identity trees with n internal (non-leaf) nodes.
%C A108530 Also for n>0, rooted trees with n nodes and 2-colored internal nodes. Black nodes correspond to nodes with a leaf child; white nodes correspond to those without one.
%H A108530 Alois P. Heinz, <a href="/A108530/b108530.txt">Table of n, a(n) for n = 0..1000</a>
%H A108530 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%H A108530 <a href="/index/Ro#rooted">Index entries for sequences related to rooted trees</a>
%F A108530 Shifts left and halves under WEIGH transform.
%F A108530 a(n) ~ c * d^n / n^(3/2), where d = 4.1516890102085520777311008746639624... and c = 0.3329810927479684511418598248... - _Vaclav Kotesovec_, Feb 28 2014
%p A108530 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A108530       add(binomial(a(i$2), j)*b(n-i*j, i-1), j=0..n/i)))
%p A108530     end:
%p A108530 a:= n-> `if`(n<2, 1, 2*b(n-1,n-1)):
%p A108530 seq(a(n), n=0..30);  # _Alois P. Heinz_, May 20 2013
%t A108530 b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, Sum[Binomial[a[i], j]*b[n - i*j, i-1], {j, 0, n/i}]]];
%t A108530 a[n_] := If[n<2, 1, 2*b[n-1, n-1]];
%t A108530 Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, Mar 01 2016, after _Alois P. Heinz_ *)
%Y A108530 Cf. A004111, A004113, A108531, A108532.
%K A108530 nonn,eigen
%O A108530 0,3
%A A108530 _Christian G. Bower_, Jun 07 2005