%I #12 Dec 09 2020 15:55:18
%S 1,0,1,0,1,0,0,1,1,0,0,1,1,0,0,0,1,3,0,0,0,0,1,3,0,0,0,0,0,1,6,1,0,0,
%T 0,0,0,1,7,1,0,0,0,0,0,0,1,11,4,0,0,0,0,0,0,0,1,13,6,0,0,0,0,0,0,0,0,
%U 1,20,16,0,0,0,0,0,0,0,0,0,1,23,23,0,0,0
%N Regular triangle where T(n,k) is the number of unlabeled series-reduced rooted trees with n leaves in which every leaf is at height k.
%H Andrew Howroyd, <a href="/A320179/b320179.txt">Table of n, a(n) for n = 1..1275</a> (rows 1..50)
%e Triangle begins:
%e 1
%e 0 1
%e 0 1 0
%e 0 1 1 0
%e 0 1 1 0 0
%e 0 1 3 0 0 0
%e 0 1 3 0 0 0 0
%e 0 1 6 1 0 0 0 0
%e 0 1 7 1 0 0 0 0 0
%e 0 1 11 4 0 0 0 0 0 0
%e 0 1 13 6 0 0 0 0 0 0 0
%e 0 1 20 16 0 0 0 0 0 0 0 0
%e 0 1 23 23 0 0 0 0 0 0 0 0 0
%e 0 1 33 46 0 0 0 0 0 0 0 0 0 0
%e The T(10,3) = 4 rooted trees:
%e (((oo)(oo))((oo)(oooo)))
%e (((oo)(oo))((ooo)(ooo)))
%e (((oo)(ooo))((oo)(ooo)))
%e (((oo)(oo))((oo)(oo)(oo)))
%t qurt[n_]:=If[n==1,{{}},Join@@Table[Union[Sort/@Tuples[qurt/@ptn]],{ptn,Select[IntegerPartitions[n],Length[#]>1&]}]];
%t Table[Length[Select[qurt[n],SameQ[##,k]&@@Length/@Position[#,{}]&]],{n,14},{k,0,n-1}]
%o (PARI) EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
%o T(n)={my(u=vector(n), v=vector(n), h=1); u[1]=1; while(u, v+=u*h; h*=x; u=EulerT(u)-u); vector(n, n, Vecrev(v[n], n))}
%o { my(A=T(15)); for(n=1, #A, print(A[n])) } \\ _Andrew Howroyd_, Dec 09 2020
%Y Row sums are A120803. Third column is A083751. An irregular version is A320221.
%Y Cf. A000669, A001678, A048816, A079500, A119262, A244925, A316655, A319312.
%Y Cf. A316624, A320154, A320155, A320160, A320172, A320173.
%K nonn,tabl
%O 1,18
%A _Gus Wiseman_, Oct 07 2018