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

1,18

Links

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

Example

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)))

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}]

Prog

(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

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.

Sequence in context: A111417 A007271 A327449 * A035656 A325675 A279948

Adjacent sequences: A320176 A320177 A320178 * A320180 A320181 A320182

Keyword

nonn,tabl

Author

Gus Wiseman, Oct 07 2018

Status

approved