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%I #14 Feb 05 2015 05:59:45
%S 0,0,0,1,2,3,4,6,9,16,28,51,91,169,312,588,1113,2125,4068,7844,15174,
%T 29505,57553,112680,221224,435637,859958,1701775,3374860,6706722,
%U 13352832,26632446,53205674,106458153,213317105,428021186,859924107,1729741358,3483357290
%N Number of rooted trees with n nodes having some subtrees replaced by cycles such that no leaf nodes are left over.
%H Alois P. Heinz, <a href="/A213682/b213682.txt">Table of n, a(n) for n = 0..1000</a>
%F a(n) ~ c * d^n / n^(3/2), where d = 2.097576205396085315047533452..., c = 0.47373451480298310710830384... . - _Vaclav Kotesovec_, Sep 07 2014
%e ..............................................
%e : o : o o : o o o :
%e : / \ : / \ | : / \ | | :
%e : o---o : o o o : o o o o :
%e : : \ / / \ : | | / \ | :
%e : : o o---o : o---o o o o :
%e : : : \ / / \ :
%e : n=3 : n=4 : n=5 o o---o :
%e :.......:..............:.....................:
%e : o o o o :
%e : / \ | | | :
%e : o o o o o :
%e : | | / \ | | :
%e : o o o o o o :
%e : \ / | | / \ | :
%e : o o---o o o o :
%e : \ / / \ :
%e : n=6 o o---o :
%e :............................:....................
%e : o o o o o o :
%e : / \ | | | | / \ :
%e : o o o o o o / \ :
%e : | | / \ | | | o o :
%e : o o o o o o o / \ / \ :
%e : | | | | / \ | | o---o o---o :
%e : o---o o o o o o o :
%e : \ / | | / \ | :
%e : o o---o o o o :
%e : \ / / \ :
%e : n=7 o o---o :
%e :................................................:
%p b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p add(binomial(a(i)+j-1, j)*b(n-i*j, i-1), j=0..n/i)))
%p end:
%p a:= n-> `if`(n<3, 0, 1+b(n-1, n-1)):
%p seq(a(n), n=0..40);
%t 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}]] // FullSimplify]; a[n_] := If[n<3, 0, 1+b[n-1, n-1]]; Table[a[n], {n, 0, 40}] (* _Jean-François Alcover_, Feb 05 2015, after _Alois P. Heinz_ *)
%Y Cf. A000081, A213674, A213683.
%K nonn
%O 0,5
%A _Alois P. Heinz_, Mar 04 2013