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%I #21 Jun 11 2021 13:37:20
%S 1,1,1,2,5,12,37,116,412,1526,5995,24284,101619,434402,1893983,
%T 8385952,37637803,170871486,783611214,3625508762,16906577279,
%U 79395295122,375217952457,1783447124452,8521191260092,40907997006020,197248252895597,954915026282162
%N Number of Greg trees with n black nodes.
%C 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 Alois P. Heinz, <a href="/A052302/b052302.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Tra#trees">Index entries for sequences related to trees</a>
%F G.f.: 1 + B(x) - B(x)^2 where B(x) is g.f. of A052300.
%p b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p add(binomial(g(i)+j-1, j)*b(n-i*j, i-1), j=0..n/i)))
%p end:
%p g:= n-> `if`(n<1, 0, b(n-1$2)+b(n, n-1)):
%p a:= n-> `if`(n=0, 1, g(n)-add(g(j)*g(n-j), j=0..n)):
%p seq(a(n), n=0..40); # _Alois P. Heinz_, Jun 22 2018
%t b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0,
%t Sum[Binomial[g[i] + j - 1, j]*b[n - i*j, i - 1], {j, 0, n/i}]]];
%t g[n_] := If[n < 1, 0, b[n - 1, n - 1] + b[n, n - 1]];
%t a[n_] := If[n == 0, 1, g[n] - Sum[g[j]*g[n - j], {j, 0, n}]];
%t a /@ Range[0, 40] (* _Jean-François Alcover_, Jun 11 2021, after _Alois P. Heinz_ *)
%Y Cf. A005263, A005264, A048159, A048160, A052300, A052301, A052303.
%K nonn
%O 0,4
%A _Christian G. Bower_, Nov 15 1999
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