A317098 - OEIS

%I #10 Aug 20 2018 20:52:32

%S 1,1,2,5,12,31,80,214,576,1595,4448,12625,36146,104662,305251,897417,

%T 2654072,7895394,23601441,70871693,213660535,646484951,1962507610,

%U 5975425743,18243789556,55841543003,171320324878,526738779846,1622739134873,5008518981670

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

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

%H Andrew Howroyd, <a href="/A317098/b317098.txt">Table of n, a(n) for n = 1..200</a>

%e The a(5) = 12 trees:

%e   (o(o(o(oo))))

%e   (o(o(ooo)))

%e   (o((oo)(oo)))

%e   (o(oo(oo)))

%e   (o(oooo))

%e   ((oo)(o(oo)))

%e   ((oo)(ooo))

%e   (oo(o(oo)))

%e   (oo(ooo))

%e   (o(oo)(oo))

%e   (ooo(oo))

%e   (ooooo)

%t semisameQ[u_]:=Length[Union[u]]<=2;

%t nms[n_]:=nms[n]=If[n==1,{{1}},Join@@Table[Select[Union[Sort/@Tuples[nms/@ptn]],semisameQ],{ptn,Rest[IntegerPartitions[n]]}]];

%t Table[Length[nms[n]],{n,10}]

%o (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

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

%K nonn

%O 1,3

%A _Gus Wiseman_, Aug 01 2018

%E Terms a(21) and beyond from _Andrew Howroyd_, Aug 19 2018