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A303023 Number of anti-binary (no binary branchings) unlabeled rooted trees with n nodes.

%I A303023 #23 May 17 2021 02:40:07
%S A303023 1,1,1,2,4,8,16,32,66,139,297,642,1404,3097,6888,15428,34770,78785,
%T A303023 179397,410264,941935,2170275,5016604,11630024,27034824,63000261,
%U A303023 147148341,344419767,807746487,1897829065,4466643367,10529301944,24858143953,58769113863
%N A303023 Number of anti-binary (no binary branchings) unlabeled rooted trees with n nodes.
%H A303023 Alois P. Heinz, <a href="/A303023/b303023.txt">Table of n, a(n) for n = 1..2000</a>
%e A303023 The a(6) = 8 rooted trees:
%e A303023   (((((o)))))
%e A303023   (((ooo)))
%e A303023   ((oo(o)))
%e A303023   (oo((o)))
%e A303023   (o(o)(o))
%e A303023   ((oooo))
%e A303023   (ooo(o))
%e A303023   (ooooo)
%p A303023 b:= proc(n, i, t) option remember; `if`(n=0, `if`(t=1, 0, 1), `if`(i<1, 0,
%p A303023       add(b(n-i*j, i-1, max(0, t-j))*binomial(a(i)+j-1, j), j=0..n/i)))
%p A303023     end:
%p A303023 a:= n-> `if`(n<2, n, b(n-1$2, 3)):
%p A303023 seq(a(n), n=1..50);  # _Alois P. Heinz_, Aug 27 2018
%t A303023 burt[n_]:=burt[n]=If[n==1,{{}},Join@@Table[Union[Sort/@Tuples[burt/@c]],{c,Select[IntegerPartitions[n-1],Length[#]!=2&]}]];
%t A303023 Table[Length[burt[n]],{n,20}]
%t A303023 (* Second program: *)
%t A303023 b[n_, i_, t_] := b[n, i, t] = If[n == 0, If[t == 1, 0, 1], If[i < 1, 0, Sum[b[n-i*j, i-1, Max[0, t-j]]*Binomial[a[i]+j-1, j], {j, 0, n/i}]]];
%t A303023 a[n_] := If[n < 2, n, b[n-1, n-1, 3]];
%t A303023 Array[a, 50] (* _Jean-François Alcover_, May 16 2021, after _Alois P. Heinz_ *)
%Y A303023 Cf. A000081, A000598, A001190, A001678, A102403, A292050, A298204, A298422.
%Y A303023 Cf. A303022, A303024, A303025, A303026, A303027, A318520.
%K A303023 nonn
%O A303023 1,4
%A A303023 _Gus Wiseman_, Aug 15 2018
%E A303023 a(24)-a(34) from _Alois P. Heinz_, Aug 27 2018