%I #25 Mar 01 2016 04:46:08
%S 1,1,2,5,14,43,138,455,1540,5305,18546,65616,234546,845683,3072350,
%T 11235393,41326470,152793376,567518950,2116666670,7924062430,
%U 29765741831,112157686170,423809991041,1605622028100,6097575361683,23207825593664,88512641860558
%N Number of asymmetric rooted Greg trees.
%C A rooted Greg tree can be described as a rooted tree with 2-colored nodes where only the black nodes are counted and the white nodes have at least 2 children.
%H Alois P. Heinz, <a href="/A052301/b052301.txt">Table of n, a(n) for n = 1..1000</a>
%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%H <a href="/index/Ro#rooted">Index entries for sequences related to rooted trees</a>
%F Satisfies a = WEIGH(a) + SHIFT_RIGHT(WEIGH(a)) - a.
%F a(n) ~ c * d^n / n^(3/2), where d = 4.0278584853545190803008179085023154..., c = 0.14959176868229550510957320468... . - _Vaclav Kotesovec_, Sep 12 2014
%p b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p add(binomial(a(i), j)*b(n-i*j, i-1), j=0..n/i)))
%p end:
%p a:= n-> `if`(n<1, 1, b(n-1$2)) +b(n, n-1):
%p seq(a(n), n=1..40); # _Alois P. Heinz_, Jul 06 2014
%t 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 a[n_] := If[n<1, 1, b[n-1, n-1]] + b[n, n-1];
%t Table[a[n], {n, 1, 40}] (* _Jean-François Alcover_, Mar 01 2016, after _Alois P. Heinz_ *)
%Y Essentially the same as A031148. Cf. A005263, A005264, A048159, A048160, A052300-A052303.
%K nonn,eigen
%O 1,3
%A _Christian G. Bower_, Nov 15 1999