|
|
A318611
|
|
Number of series-reduced powerful rooted trees with n nodes.
|
|
5
|
|
|
1, 0, 1, 1, 1, 1, 2, 1, 3, 3, 4, 4, 8, 5, 11, 10, 14, 14, 24, 18, 34, 32, 46, 45, 72, 60, 103, 96, 138, 137, 212, 184, 296, 282, 403, 397, 591, 539, 830, 798, 1125, 1119, 1624, 1519, 2253, 2195, 3067, 3056, 4341, 4158, 6004, 5897, 8145, 8164, 11397, 11090
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
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
|
|
|
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)):
|
|
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]];
|
|
CROSSREFS
|
Cf. A000081, A001190, A001678, A001694, A004111, A167865, A291636, A317102, A317705, A317707, A318612, A318691.
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|