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

1, 0, 0, 1, 1, 1, 2, 3, 4, 7, 11, 17, 28, 46, 74, 123, 205, 341, 571, 964, 1629, 2764, 4707, 8040, 13766, 23639, 40681, 70163, 121256, 209960, 364168, 632694, 1100906, 1918375, 3347346, 5848271, 10229977, 17915018, 31407088, 55116661, 96818589, 170229939

Offset: 1

Author

Gus Wiseman, Aug 15 2018

Keywords

• nonn

Examples

			The a(10) = 7 rooted trees:
  (oo(oo(ooo)))
  (o(ooo)(ooo))
  (oo(oooooo))
  (ooo(ooooo))
  (oooo(oooo))
  (ooooo(ooo))
  (ooooooooo)
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..2000

Crossrefs

Cf. A000081, A000598, A001190, A001678, A102403, A298204, A298422.

Cf. A303022, A303023, A303024, A303026, A303027.

Programs

Maple

b:= proc(n, i, t) option remember; `if`(n=0, `if`(t=0, 1, 0), `if`(i<1, 0,
      add(b(n-i*j, i-1, max(0, t-j))*binomial(a(i)+j-1, j), j=0..n/i)))
    end:
a:= n-> `if`(n<2, n, b(n-1$2, 3)):
seq(a(n), n=1..50);  # Alois P. Heinz, Aug 27 2018

Mathematica

zurt[n_]:=zurt[n]=If[n==1,{{}},Join@@Table[Union[Sort/@Tuples[zurt/@c]],{c,Select[IntegerPartitions[n-1],Length[#]>2&]}]];
Table[Length[zurt[n]],{n,20}]
(* Second program: *)
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}]]];
a[n_] :=  If[n < 2, n, b[n-1, n-1, 3]];
Array[a, 50] (* Jean-François Alcover, May 17 2021, after Alois P. Heinz *)

Extensions

a(36)-a(42) from Alois P. Heinz, Aug 27 2018