A317098 Number of series-reduced rooted trees with n unlabeled leaves where the number of distinct branches under each node is <= 2.

1, 1, 2, 5, 12, 31, 80, 214, 576, 1595, 4448, 12625, 36146, 104662, 305251, 897417, 2654072, 7895394, 23601441, 70871693, 213660535, 646484951, 1962507610, 5975425743, 18243789556, 55841543003, 171320324878, 526738779846, 1622739134873, 5008518981670

Offset: 1

Author

Gus Wiseman, Aug 01 2018

Keywords

• nonn

Comments

There can be more than two branches as long as there are not three distinct branches.

Examples

			The a(5) = 12 trees:
  (o(o(o(oo))))
  (o(o(ooo)))
  (o((oo)(oo)))
  (o(oo(oo)))
  (o(oooo))
  ((oo)(o(oo)))
  ((oo)(ooo))
  (oo(o(oo)))
  (oo(ooo))
  (o(oo)(oo))
  (ooo(oo))
  (ooooo)
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..200

Crossrefs

Cf. A000081, A000598, A001190, A055277, A111299, A292050, A298204, A301344, A317097.

Programs

Mathematica

semisameQ[u_]:=Length[Union[u]]<=2;
nms[n_]:=nms[n]=If[n==1,{{1}},Join@@Table[Select[Union[Sort/@Tuples[nms/@ptn]],semisameQ],{ptn,Rest[IntegerPartitions[n]]}]];
Table[Length[nms[n]],{n,10}]

PARI

seq(n)={my(v=vector(n)); v[1]=1; for(n=2, n, v[n]=sum(k=1, n-1, sumdiv(k, d, v[d])*sumdiv(n-k, d, v[d])/2) + sumdiv(n, d, v[n/d]*(1 - (d-1)/2)) ); v} \\ Andrew Howroyd, Aug 19 2018

Extensions

Terms a(21) and beyond from Andrew Howroyd, Aug 19 2018