%I #20 Feb 25 2025 10:36:06
%S 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,
%T 13,16,14,22,22,24,21,24,28,14,32,15,42,20,60,27,84,44,100,59,113,74,
%U 116,85,110,97,96,113,106,149,147,234,235,377,380,580,576,838
%N 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.
%C 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.
%H Robert G. Wilson v, <a href="/A297791/b297791.txt">Table of n, a(n) for n = 1..84</a>
%e The a(13) = 5 trees: (((oo)(oo))(oooo)), ((ooooo)(ooooo)), ((ooo)(ooo)(ooo)), ((oo)(oo)(oo)(oo)), (oooooooooooo).
%t 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]];
%t Table[Length[alltim[n]],{n,20}]
%o (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
%Y Cf. A000081, A001190, A001678, A006241, A032305, A289078, A289079, A291441, A291443.
%K nonn,changed
%O 1,7
%A _Gus Wiseman_, Jan 06 2018
%E a(51) onward from _Robert G. Wilson v_, Jan 07 2018