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A303025 Number of series-reduced anti-binary (no unary or binary branchings) unlabeled rooted trees with n nodes.

%I A303025 #19 May 17 2021 15:05:08
%S A303025 1,0,0,1,1,1,2,3,4,7,11,17,28,46,74,123,205,341,571,964,1629,2764,
%T A303025 4707,8040,13766,23639,40681,70163,121256,209960,364168,632694,
%U A303025 1100906,1918375,3347346,5848271,10229977,17915018,31407088,55116661,96818589,170229939
%N A303025 Number of series-reduced anti-binary (no unary or binary branchings) unlabeled rooted trees with n nodes.
%H A303025 Alois P. Heinz, <a href="/A303025/b303025.txt">Table of n, a(n) for n = 1..2000</a>
%e A303025 The a(10) = 7 rooted trees:
%e A303025   (oo(oo(ooo)))
%e A303025   (o(ooo)(ooo))
%e A303025   (oo(oooooo))
%e A303025   (ooo(ooooo))
%e A303025   (oooo(oooo))
%e A303025   (ooooo(ooo))
%e A303025   (ooooooooo)
%p A303025 b:= proc(n, i, t) option remember; `if`(n=0, `if`(t=0, 1, 0), `if`(i<1, 0,
%p A303025       add(b(n-i*j, i-1, max(0, t-j))*binomial(a(i)+j-1, j), j=0..n/i)))
%p A303025     end:
%p A303025 a:= n-> `if`(n<2, n, b(n-1$2, 3)):
%p A303025 seq(a(n), n=1..50);  # _Alois P. Heinz_, Aug 27 2018
%t A303025 zurt[n_]:=zurt[n]=If[n==1,{{}},Join@@Table[Union[Sort/@Tuples[zurt/@c]],{c,Select[IntegerPartitions[n-1],Length[#]>2&]}]];
%t A303025 Table[Length[zurt[n]],{n,20}]
%t A303025 (* Second program: *)
%t A303025 b[n_, i_, t_] := b[n, i, t] = If[n == 0, If[t == 0, 1, 0], 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 A303025 a[n_] :=  If[n < 2, n, b[n-1, n-1, 3]];
%t A303025 Array[a, 50] (* _Jean-François Alcover_, May 17 2021, after _Alois P. Heinz_ *)
%Y A303025 Cf. A000081, A000598, A001190, A001678, A102403, A298204, A298422.
%Y A303025 Cf. A303022, A303023, A303024, A303026, A303027.
%K A303025 nonn
%O A303025 1,7
%A A303025 _Gus Wiseman_, Aug 15 2018
%E A303025 a(36)-a(42) from _Alois P. Heinz_, Aug 27 2018