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A303025
Number of series-reduced anti-binary (no unary or binary branchings) unlabeled rooted trees with n nodes.
6
1, 0, 0, 1, 1, 1, 2, 3, 4, 7, 11, 17, 28, 46, 74, 123, 205, 341, 571, 964, 1629, 2764, 4707, 8040, 13766, 23639, 40681, 70163, 121256, 209960, 364168, 632694, 1100906, 1918375, 3347346, 5848271, 10229977, 17915018, 31407088, 55116661, 96818589, 170229939
OFFSET
1,7
LINKS
EXAMPLE
The a(10) = 7 rooted trees:
(oo(oo(ooo)))
(o(ooo)(ooo))
(oo(oooooo))
(ooo(ooooo))
(oooo(oooo))
(ooooo(ooo))
(ooooooooo)
MAPLE
b:= proc(n, i, t) option remember; `if`(n=0, `if`(t=0, 1, 0), `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
zurt[n_]:=zurt[n]=If[n==1, {{}}, Join@@Table[Union[Sort/@Tuples[zurt/@c]], {c, Select[IntegerPartitions[n-1], Length[#]>2&]}]];
Table[Length[zurt[n]], {n, 20}]
(* Second program: *)
b[n_, i_, t_] := b[n, i, t] = If[n == 0, If[t == 0, 1, 0], 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 17 2021, after Alois P. Heinz *)
KEYWORD
nonn
AUTHOR
Gus Wiseman, Aug 15 2018
EXTENSIONS
a(36)-a(42) from Alois P. Heinz, Aug 27 2018
STATUS
approved