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A052303 Number of asymmetric Greg trees.

%I A052303 #17 Apr 28 2020 07:33:16
%S A052303 1,1,0,0,0,0,1,4,12,42,137,452,1491,4994,16831,57408,197400,685008,
%T A052303 2395310,8437830,29917709,106724174,382807427,1380058180,4998370015,
%U A052303 18181067670,66393725289,243347195594,894959868983,3301849331598,12217869541117,45335177297876
%N A052303 Number of asymmetric Greg trees.
%C A052303 A Greg tree can be described as a tree with 2-colored nodes where only the black nodes are counted and the white nodes are of degree at least 3.
%H A052303 Alois P. Heinz, <a href="/A052303/b052303.txt">Table of n, a(n) for n = 0..1668</a>
%H A052303 <a href="/index/Tra#trees">Index entries for sequences related to trees</a>
%F A052303 G.f.: 1+B(x)-B(x)^2 where B(x) is g.f. of A052301.
%p A052303 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A052303       add(binomial(g(i), j)*b(n-i*j, i-1), j=0..n/i)))
%p A052303     end:
%p A052303 g:= n-> `if`(n<1, 0, b(n-1$2)+b(n, n-1)) :
%p A052303 a:= n-> `if`(n=0, 1, g(n)-add(g(j)*g(n-j), j=0..n)):
%p A052303 seq(a(n), n=0..40);  # _Alois P. Heinz_, Jun 22 2018
%t A052303 b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, Sum[Binomial[g[i], j] b[n - i j, i - 1], {j, 0, n/i}]]];
%t A052303 g[n_] := If[n < 1, 0, b[n - 1, n - 1] + b[n, n - 1]];
%t A052303 a[n_] := If[n == 0, 1, g[n] - Sum[g[j] g[n - j], {j, 0, n}]];
%t A052303 a /@ Range[0, 40] (* _Jean-François Alcover_, Apr 28 2020, after _Alois P. Heinz_ *)
%Y A052303 Cf. A005263, A005264, A048159, A048160, A052300-A052302.
%K A052303 nonn
%O A052303 0,8
%A A052303 _Christian G. Bower_, Nov 15 1999