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A213920 Number of rooted trees with n nodes such that no more than two subtrees corresponding to children of any node have the same number of nodes.

%I A213920 #22 Jun 04 2019 04:43:26
%S A213920 0,1,1,2,3,7,15,34,79,190,457,1132,2823,7126,18136,46541,120103,
%T A213920 312109,815012,2137755,5632399,14895684,39519502,105198371,280815067,
%U A213920 751490363,2016142768,5420945437,14604580683,39425557103,106618273626,288792927325,783516425820
%N A213920 Number of rooted trees with n nodes such that no more than two subtrees corresponding to children of any node have the same number of nodes.
%C A213920 Coincides with A248869 up to a(9) = 190.
%C A213920 a(n+1)/a(n) tends to 2.845331... - _Vaclav Kotesovec_, Jun 04 2019
%H A213920 Alois P. Heinz, <a href="/A213920/b213920.txt">Table of n, a(n) for n = 0..2213</a>
%e A213920 :  o  :  o  :    o   o  :    o     o   o  :
%e A213920 :     :  |  :   / \  |  :    |    / \  |  :
%e A213920 :     :  o  :  o   o o  :    o   o   o o  :
%e A213920 :     :     :        |  :   / \  |     |  :
%e A213920 :     :     :        o  :  o   o o     o  :
%e A213920 :     :     :           :              |  :
%e A213920 : n=1 : n=2 :  n=3      :  n=4         o  :
%e A213920 :.....:.....:...........:.................:
%e A213920 :   o     o       o     o     o     o   o :
%e A213920 :   |     |      / \   / \   / \   /|\  | :
%e A213920 :   o     o     o   o o   o o   o o o o o :
%e A213920 :   |    / \   / \    |     |   | |     | :
%e A213920 :   o   o   o o   o   o     o   o o     o :
%e A213920 :  / \  |             |                 | :
%e A213920 : o   o o             o                 o :
%e A213920 :                                       | :
%e A213920 : n=5                                   o :
%e A213920 :.........................................:
%p A213920 g:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
%p A213920       binomial(g((i-1)$2)+j-1, j)*g(n-i*j, i-1), j=0..min(2, n/i))))
%p A213920     end:
%p A213920 a:= n-> g((n-1)$2):
%p A213920 seq(a(n), n=0..40);
%t A213920 g[n_, i_] := g[n, i] = If[n==0, 1, If[i<1, 0, Sum[Binomial[g[i-1, i-1]+j-1, j]*g[n-i*j, i-1], {j, 0, Min[2, n/i]}]]]; a[n_] := g[n-1, n-1]; Table[ a[n], {n, 0, 40}] (* _Jean-François Alcover_, Feb 21 2017, translated from Maple *)
%Y A213920 Cf. A000081, A032305, A248869.
%Y A213920 Column k=2 of A318753.
%K A213920 nonn
%O A213920 0,4
%A A213920 _Alois P. Heinz_, Mar 05 2013