|
|
A317098
|
|
Number of series-reduced rooted trees with n unlabeled leaves where the number of distinct branches under each node is <= 2.
|
|
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
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,3
|
|
COMMENTS
|
There can be more than two branches as long as there are not three distinct branches.
|
|
LINKS
|
|
|
EXAMPLE
|
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)
|
|
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}]
|
|
PROG
|
(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
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|