|
|
A358587
|
|
Number of n-node rooted trees of height equal to the number of internal (non-leaf) nodes.
|
|
18
|
|
|
0, 0, 0, 0, 1, 4, 14, 41, 111, 282, 688, 1627, 3761, 8540, 19122, 42333, 92851, 202078, 436916, 939359, 2009781, 4281696, 9087670, 19223905, 40544951, 85284194, 178956984, 374691171, 782936761, 1632982372, 3400182458, 7068800357, 14674471611, 30422685030
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,6
|
|
LINKS
|
|
|
FORMULA
|
a(n) = 5*a(n-1) - 7*a(n-2) - a(n-3) + 8*a(n-4) - 4*a(n-5) for n > 7.
G.f.: x^5*(x^2 - x + 1)/((x - 1)^2*(x + 1)*(2*x - 1)^2). (End)
|
|
EXAMPLE
|
The a(5) = 1 through a(7) = 14 trees:
((o)(o)) ((o)(oo)) ((o)(ooo))
(o(o)(o)) ((oo)(oo))
(((o)(o))) (o(o)(oo))
((o)((o))) (oo(o)(o))
(((o))(oo))
(((o)(oo)))
((o)((oo)))
((o)(o(o)))
((o(o)(o)))
(o((o)(o)))
(o(o)((o)))
((((o)(o))))
(((o)((o))))
((o)(((o))))
|
|
MATHEMATICA
|
art[n_]:=If[n==1, {{}}, Join@@Table[Select[Tuples[art/@c], OrderedQ], {c, Join@@Permutations/@IntegerPartitions[n-1]}]];
Table[Length[Select[art[n], Count[#, _[__], {0, Infinity}]==Depth[#]-1&]], {n, 1, 10}]
|
|
PROG
|
(PARI) \\ Needs R(n, f) defined in A358589.
seq(n) = {Vec(R(n, (h, p)->polcoef(subst(p, x, x/y), -h, y)), -n)} \\ Andrew Howroyd, Jan 01 2023
|
|
CROSSREFS
|
For leaves instead of height we have A185650 aerated, ranked by A358578.
A358575 counts rooted trees by nodes and internal nodes, ordered A090181.
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|