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A303023
Number of anti-binary (no binary branchings) unlabeled rooted trees with n nodes.
7
1, 1, 1, 2, 4, 8, 16, 32, 66, 139, 297, 642, 1404, 3097, 6888, 15428, 34770, 78785, 179397, 410264, 941935, 2170275, 5016604, 11630024, 27034824, 63000261, 147148341, 344419767, 807746487, 1897829065, 4466643367, 10529301944, 24858143953, 58769113863
OFFSET
1,4
LINKS
EXAMPLE
The a(6) = 8 rooted trees:
(((((o)))))
(((ooo)))
((oo(o)))
(oo((o)))
(o(o)(o))
((oooo))
(ooo(o))
(ooooo)
MAPLE
b:= proc(n, i, t) option remember; `if`(n=0, `if`(t=1, 0, 1), `if`(i<1, 0,
add(b(n-i*j, i-1, max(0, t-j))*binomial(a(i)+j-1, j), j=0..n/i)))
end:
a:= n-> `if`(n<2, n, b(n-1$2, 3)):
seq(a(n), n=1..50); # Alois P. Heinz, Aug 27 2018
MATHEMATICA
burt[n_]:=burt[n]=If[n==1, {{}}, Join@@Table[Union[Sort/@Tuples[burt/@c]], {c, Select[IntegerPartitions[n-1], Length[#]!=2&]}]];
Table[Length[burt[n]], {n, 20}]
(* Second program: *)
b[n_, i_, t_] := b[n, i, t] = If[n == 0, If[t == 1, 0, 1], If[i < 1, 0, Sum[b[n-i*j, i-1, Max[0, t-j]]*Binomial[a[i]+j-1, j], {j, 0, n/i}]]];
a[n_] := If[n < 2, n, b[n-1, n-1, 3]];
Array[a, 50] (* Jean-François Alcover, May 16 2021, after Alois P. Heinz *)
KEYWORD
nonn
AUTHOR
Gus Wiseman, Aug 15 2018
EXTENSIONS
a(24)-a(34) from Alois P. Heinz, Aug 27 2018
STATUS
approved