OFFSET
1,7
COMMENTS
A series-reduced rooted tree is powerful if either it is a single node, or the branches of the root all appear with multiplicities greater than 1 and are themselves series-reduced powerful rooted trees.
LINKS
Alois P. Heinz, Table of n, a(n) for n = 1..8000
EXAMPLE
The a(13) = 8 series-reduced powerful rooted trees:
((oo)(oo)(oo)(oo))
((ooo)(ooo)(ooo))
(ooo(oo)(oo)(oo))
((ooooo)(ooooo))
(oo(oooo)(oooo))
(oooo(ooo)(ooo))
(oooooo(oo)(oo))
(oooooooooooo)
MAPLE
h:= proc(n, k, t) option remember; `if`(k=0, binomial(n+t, t),
`if`(n=0, 0, add(h(n-1, k-j, t+1), j=2..k)))
end:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
add(b(n-i*j, i-1)*h(a(i), j, 0), j=0..n/i)))
end:
a:= n-> `if`(n<2, n, b(n-1$2)):
seq(a(n), n=1..60); # Alois P. Heinz, Aug 31 2018
MATHEMATICA
purt[n_]:=purt[n]=If[n==1, {{}}, Join@@Table[Select[Union[Sort/@Tuples[purt/@ptn]], Min@@Length/@Split[#]>1&], {ptn, IntegerPartitions[n-1]}]];
Table[Length[purt[n]], {n, 20}]
(* Second program: *)
h[n_, k_, t_] := h[n, k, t] = If[k == 0, Binomial[n + t, t],
If[n == 0, 0, Sum[h[n - 1, k - j, t + 1], {j, 2, k}]]];
b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0,
Sum[b[n - i*j, i - 1]*h[a[i], j, 0], {j, 0, n/i}]]];
a[n_] := If[n < 2, n, b[n - 1, n - 1]];
Array[a, 60] (* Jean-François Alcover, May 19 2021, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, Aug 30 2018
EXTENSIONS
a(41)-a(56) from Alois P. Heinz, Aug 31 2018
STATUS
approved