|
|
A358728
|
|
Number of n-node rooted trees whose node-height is less than their number of leaves.
|
|
3
|
|
|
0, 0, 0, 1, 1, 5, 10, 30, 76, 219, 582, 1662, 4614, 13080, 36903, 105098, 298689, 852734, 2434660, 6964349, 19931147, 57100177, 163647811, 469290004, 1346225668, 3863239150, 11089085961, 31838349956, 91430943515, 262615909503, 754439588007, 2167711283560
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,6
|
|
COMMENTS
|
Node-height is the number of nodes in the longest path from root to leaf.
|
|
LINKS
|
|
|
EXAMPLE
|
The a(1) = 0 through a(7) = 10 trees:
. . . (ooo) (oooo) (ooooo) (oooooo)
((oooo)) ((ooooo))
(o(ooo)) (o(oooo))
(oo(oo)) (oo(ooo))
(ooo(o)) (ooo(oo))
(oooo(o))
((o)(ooo))
((oo)(oo))
(o(o)(oo))
(oo(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], Depth[#]-1<Count[#, {}, {-2}]&]], {n, 1, 10}]
|
|
PROG
|
(PARI) \\ Needs R(n, f) defined in A358589.
seq(n) = {Vec(R(n, (h, p)->sum(j=h+1, n-1, polcoef(p, j, y))), -n)} \\ Andrew Howroyd, Jan 01 2023
|
|
CROSSREFS
|
For internals instead of node-height we have A358581, ordered A358585.
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|