A320179 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.

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, 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, 1, 20, 16, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 23, 23, 0, 0, 0

Offset: 1

Author

Gus Wiseman, Oct 07 2018

Keywords

• nonn
• tabl

Examples

			Triangle begins:
  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  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  1 20 16  0  0  0  0  0  0  0  0
  0  1 23 23  0  0  0  0  0  0  0  0  0
  0  1 33 46  0  0  0  0  0  0  0  0  0  0
The T(10,3) = 4 rooted trees:
   (((oo)(oo))((oo)(oooo)))
   (((oo)(oo))((ooo)(ooo)))
   (((oo)(ooo))((oo)(ooo)))
  (((oo)(oo))((oo)(oo)(oo)))
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..1275 (rows 1..50)

Crossrefs

Row sums are A120803. Third column is A083751. An irregular version is A320221.

Cf. A000669, A001678, A048816, A079500, A119262, A244925, A316655, A319312.

Cf. A316624, A320154, A320155, A320160, A320172, A320173.

Programs

Mathematica

qurt[n_]:=If[n==1,{{}},Join@@Table[Union[Sort/@Tuples[qurt/@ptn]],{ptn,Select[IntegerPartitions[n],Length[#]>1&]}]];
Table[Length[Select[qurt[n],SameQ[##,k]&@@Length/@Position[#,{}]&]],{n,14},{k,0,n-1}]

PARI

EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
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))}
{ my(A=T(15)); for(n=1, #A, print(A[n])) } \\ Andrew Howroyd, Dec 09 2020