cp's OEIS Frontend

A320155 Number of series-reduced balanced rooted trees with n labeled leaves.

%I A320155 #21 Nov 18 2021 03:58:28
%S A320155 1,1,1,4,11,41,162,1030,7205,55522,442443,3810852,35272030,351697516,
%T A320155 3735838550,42719792640,529195988635,7128835815387,103651381499810,
%U A320155 1610812109555323,26489497655582729,457497408108551450,8248899117402701046,154624472715479106919
%N A320155 Number of series-reduced balanced rooted trees with n labeled leaves.
%C A320155 A rooted tree is series-reduced if every non-leaf node has at least two branches, and balanced if all leaves are the same distance from the root.
%H A320155 Andrew Howroyd, <a href="/A320155/b320155.txt">Table of n, a(n) for n = 1..200</a>
%F A320155 E.g.f. A(x) satisfies A(x) = x + A(exp(x)-x-1). - _Ira M. Gessel_, Nov 17 2021
%e A320155 The a(1) = 1 through a(5) = 11 rooted trees:
%e A320155   1  (12)  (123)    (1234)      (12345)
%e A320155                   ((12)(34))  ((12)(345))
%e A320155                   ((13)(24))  ((13)(245))
%e A320155                   ((14)(23))  ((14)(235))
%e A320155                               ((15)(234))
%e A320155                               ((23)(145))
%e A320155                               ((24)(135))
%e A320155                               ((25)(134))
%e A320155                               ((34)(125))
%e A320155                               ((35)(124))
%e A320155                               ((45)(123))
%t A320155 sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t A320155 phy2[labs_]:=If[Length[labs]==1,labs,Union@@Table[Sort/@Tuples[phy2/@ptn],{ptn,Select[sps[Sort[labs]],Length[#1]>1&]}]];
%t A320155 Table[Length[Select[phy2[Range[n]],SameQ@@Length/@Position[#,_Integer]&]],{n,7}]
%o A320155 (PARI) EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
%o A320155 b(n,k)={my(u=vector(n), v=vector(n)); u[1]=k; while(u, v+=u; u=EulerT(u)-u); v}
%o A320155 seq(n)={my(M=Mat(vectorv(n,k,b(n,k)))); vector(n, k, sum(i=1, k, binomial(k,i)*(-1)^(k-i)*M[i,k]))} \\ _Andrew Howroyd_, Oct 26 2018
%Y A320155 Cf. A000081, A000311, A000669, A001678, A005804, A048816, A079500, A119262, A120803, A141268, A244925, A319312.
%Y A320155 Cf. A320154, A320160, A316624, A320169, A320173, A320179.
%K A320155 nonn
%O A320155 1,4
%A A320155 _Gus Wiseman_, Oct 06 2018
%E A320155 Terms a(10) and beyond from _Andrew Howroyd_, Oct 26 2018