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

Author

Gus Wiseman, Jan 06 2018

Keywords

• nonn

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.

Examples

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

Links

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

Crossrefs

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

Programs

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}]

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(dw[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

Extensions

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