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A358582
Number of rooted trees with n nodes, most of which are not leaves.
9
0, 0, 1, 1, 5, 7, 28, 48, 176, 336, 1179, 2420, 8269, 17855, 59832, 134289, 443407, 1025685, 3346702, 7933161, 25632265, 62000170, 198670299, 488801159, 1555187172, 3882403641, 12276230777, 31034921462, 97601239282, 249471619165, 780790439063, 2015194486878
OFFSET
1,5
LINKS
FORMULA
A358581(n) + A358584(n) = A000081(n).
A358582(n) + A358583(n) = A000081(n).
a(n) = Sum_{k=1..floor((n-1)/2)} A055277(n, k). - Andrew Howroyd, Dec 30 2022
EXAMPLE
The a(3) = 1 through a(6) = 7 trees:
((o)) (((o))) (((oo))) ((((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}]<Count[#, _[__], {0, Infinity}]&]], {n, 0, 10}]
PROG
(PARI) \\ See A358584 for R(n).
seq(n) = {my(A=R(n)); vector(n, n, vecsum(Vecrev(A[n]/y)[1..(n-1)\2]))} \\ Andrew Howroyd, Dec 30 2022
CROSSREFS
For equality we have A185650 aerated, ranked by A358578.
The opposite version is A358581, non-strict A358583.
The non-strict version is A358584.
The ordered version is A358585, odd-indexed terms A065097.
A000081 counts rooted trees, ordered A000108.
A055277 counts rooted trees by nodes and leaves, ordered A001263.
A358575 counts rooted trees by nodes and internal nodes, ordered A090181.
A358589 counts square trees, ranked by A358577, ordered A358590.
Sequence in context: A268701 A317166 A126888 * A185302 A179305 A307100
KEYWORD
nonn
AUTHOR
Gus Wiseman, Nov 23 2022
EXTENSIONS
Terms a(19) and beyond from Andrew Howroyd, Dec 30 2022
STATUS
approved