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A052300 Number of rooted Greg trees.

%I A052300 #22 Jun 11 2021 12:24:59
%S A052300 1,2,6,21,78,313,1306,5653,25088,113685,523522,2443590,11533010,
%T A052300 54949539,263933658,1276652682,6213207330,30402727854,149486487326,
%U A052300 738184395770,3659440942282,18205043615467,90856842218506,454770531433586,2282393627458496,11483114908752959
%N A052300 Number of rooted Greg trees.
%C A052300 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 A052300 Alois P. Heinz, <a href="/A052300/b052300.txt">Table of n, a(n) for n = 1..1000</a>
%H A052300 <a href="/index/Ro#rooted">Index entries for sequences related to rooted trees</a>
%H A052300 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%F A052300 Satisfies a = EULER(a) + SHIFT_RIGHT(EULER(a)) - a.
%F A052300 a(n) ~ c * d^n / n^(3/2), where d = 5.33997181362574740496306748840603859910694551382103293340704... and c = 0.18146848896221859476228524468003196434835879494225205... - _Vaclav Kotesovec_, Jun 11 2021
%p A052300 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A052300       add(binomial(a(i)+j-1, j)*b(n-i*j, i-1), j=0..n/i)))
%p A052300     end:
%p A052300 a:= n-> `if`(n<1, 0, b(n-1$2)+b(n, n-1)):
%p A052300 seq(a(n), n=1..40);  # _Alois P. Heinz_, Jun 22 2018
%t A052300 b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, Sum[Binomial[a[i] + j - 1, j] b[n - i j, i - 1], {j, 0, n/i}]]];
%t A052300 a[n_] := If[n < 1, 0, b[n - 1, n - 1] + b[n, n - 1]];
%t A052300 a /@ Range[1, 40] (* _Jean-François Alcover_, Oct 02 2019, after _Alois P. Heinz_ *)
%Y A052300 Cf. A005263, A005264, A048159, A048160, A052301-A052303.
%K A052300 nonn,eigen
%O A052300 1,2
%A A052300 _Christian G. Bower_, Nov 15 1999