OFFSET
1,5
LINKS
Andrew Howroyd, Table of n, a(n) for n = 1..1000
FORMULA
From Andrew Howroyd, Jan 13 2024: (Start)
a(n) = Sum_{k=1..floor((n-1)/2)} A001263(n-1, k) for n >= 2.
a(2*n+1) = A000108(2*n)/2 for n >= 1. (End)
EXAMPLE
The a(1) = 1 through a(6) = 11 ordered trees:
o . (oo) (ooo) (oooo) (ooooo)
((o)oo) ((o)ooo)
((oo)o) ((oo)oo)
((ooo)) ((ooo)o)
(o(o)o) ((oooo))
(o(oo)) (o(o)oo)
(oo(o)) (o(oo)o)
(o(ooo))
(oo(o)o)
(oo(oo))
(ooo(o))
MATHEMATICA
aot[n_]:=If[n==1, {{}}, Join@@Table[Tuples[aot/@c], {c, Join@@Permutations/@IntegerPartitions[n-1]}]];
Table[Length[Select[aot[n], Count[#, {}, {0, Infinity}]>Count[#, _[__], {0, Infinity}]&]], {n, 10}]
PROG
(PARI) a(n) = if(n==1, 1, n--; (binomial(2*n, n)/(n+1) - if(n%2, binomial(n, (n-1)/2)^2 / n))/2) \\ Andrew Howroyd, Jan 13 2024
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, Nov 24 2022
EXTENSIONS
a(16) onwards from Andrew Howroyd, Jan 13 2024
STATUS
approved