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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.
5

%I #22 Jun 04 2019 04:43:26

%S 0,1,1,2,3,7,15,34,79,190,457,1132,2823,7126,18136,46541,120103,

%T 312109,815012,2137755,5632399,14895684,39519502,105198371,280815067,

%U 751490363,2016142768,5420945437,14604580683,39425557103,106618273626,288792927325,783516425820

%N 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 Coincides with A248869 up to a(9) = 190.

%C a(n+1)/a(n) tends to 2.845331... - _Vaclav Kotesovec_, Jun 04 2019

%H Alois P. Heinz, <a href="/A213920/b213920.txt">Table of n, a(n) for n = 0..2213</a>

%e : o : o : o o : o o o :

%e : : | : / \ | : | / \ | :

%e : : o : o o o : o o o o :

%e : : : | : / \ | | :

%e : : : o : o o o o :

%e : : : : | :

%e : n=1 : n=2 : n=3 : n=4 o :

%e :.....:.....:...........:.................:

%e : o o o o o o o :

%e : | | / \ / \ / \ /|\ | :

%e : o o o o o o o o o o o o :

%e : | / \ / \ | | | | | :

%e : o o o o o o o o o o :

%e : / \ | | | :

%e : o o o o o :

%e : | :

%e : n=5 o :

%e :.........................................:

%p g:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, add(

%p binomial(g((i-1)$2)+j-1, j)*g(n-i*j, i-1), j=0..min(2, n/i))))

%p end:

%p a:= n-> g((n-1)$2):

%p seq(a(n), n=0..40);

%t 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 Cf. A000081, A032305, A248869.

%Y Column k=2 of A318753.

%K nonn

%O 0,4

%A _Alois P. Heinz_, Mar 05 2013