A297791 Number of series-reduced leaf-balanced rooted trees with n nodes. Number of orderless same-trees with n nodes and all leaves equal to 1.

1, 0, 1, 1, 1, 1, 2, 1, 2, 2, 2, 1, 5, 1, 3, 3, 4, 3, 5, 3, 6, 4, 6, 3, 12, 3, 10, 7, 9, 6, 12, 9, 13, 16, 14, 22, 22, 24, 21, 24, 28, 14, 32, 15, 42, 20, 60, 27, 84, 44, 100, 59, 113, 74, 116, 85, 110, 97, 96, 113, 106, 149, 147, 234, 235, 377, 380, 580, 576, 838

Offset

1,7

Comments

An unlabeled rooted tree is leaf-balanced if all branches from the same root have the same number of leaves. It is series-reduced if all positive out-degrees are greater than one.

Links

Robert G. Wilson v, Table of n, a(n) for n = 1..84

Example

The a(13) = 5 trees: (((oo)(oo))(oooo)), ((ooooo)(ooooo)), ((ooo)(ooo)(ooo)), ((oo)(oo)(oo)(oo)), (oooooooooooo).

Mathematica

alltim[n_]:=alltim[n]=If[n===1, {{}}, Join@@Function[c, Select[Union[Sort/@Tuples[alltim/@c]], And[SameQ@@(Count[#, {}, {0, Infinity}]&/@#), FreeQ[#, {_}]]&]]/@IntegerPartitions[n-1]];
Table[Length[alltim[n]], {n, 20}]

Prog

(PARI) lista(nn) = my(k, r, t, u, w=vector(nn, i, vector(i))); w[1][1]=1; for(s=2, nn, fordiv(s, d, if(d<s, u=select(i->w[i][d], [d..nn]); forvec(v=vector(s/d, i, [1, #u]), if(nn>=r=1+sum(i=1, #v, u[v[i]]), k=1; t=1; for(i=2, #v, if(v[i]==v[i-1], k++, t*=binomial(w[u[v[i-1]]][d]+k-1, k); k=1)); w[r][s]+=t*binomial(w[u[v[#v]]][d]+k-1, k)), 1)))); vector(nn, i, vecsum(w[i])); \\ Jinyuan Wang, Feb 25 2025

Crossrefs

Cf. A000081, A001190, A001678, A006241, A032305, A289078, A289079, A291441, A291443.

Sequence in context: A011776 A375624 A345182 * A098965 A290087 A016443

Adjacent sequences: A297788 A297789 A297790 * A297792 A297793 A297794

Keyword

nonn

Author

Gus Wiseman, Jan 06 2018

Extensions

a(51) onward from Robert G. Wilson v, Jan 07 2018

Status

approved