A320221 Irregular 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, (n>=1, min(1,n-1) <= k <= log_2(n)).

1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 3, 1, 6, 1, 1, 7, 1, 1, 11, 4, 1, 13, 6, 1, 20, 16, 1, 23, 23, 1, 33, 46, 1, 40, 70, 1, 54, 127, 1, 1, 65, 189, 1, 1, 87, 320, 5, 1, 104, 476, 10, 1, 136, 771, 32, 1, 164, 1145, 63, 1, 209, 1795, 154, 1, 252, 2657, 304, 1, 319, 4091, 656

Offset

1,9

Links

Andrew Howroyd, Table of n, a(n) for n = 1..1154 (rows 1..200)

Example

Triangle begins:
  1
  1
  1
  1  1
  1  1
  1  3
  1  3
  1  6  1
  1  7  1
  1 11  4
  1 13  6
  1 20 16
  1 23 23
  1 33 46
  1 40 70
The T(11,3) = 6 rooted trees:
   (((oo)(oo))((oo)(ooooo)))
   (((oo)(oo))((ooo)(oooo)))
   (((oo)(ooo))((oo)(oooo)))
   (((oo)(ooo))((ooo)(ooo)))
  (((oo)(oo))((oo)(oo)(ooo)))
  (((oo)(ooo))((oo)(oo)(oo)))

Mathematica

qurt[n_]:=If[n==1, {{}}, Join@@Table[Union[Sort/@Tuples[qurt/@ptn]], {ptn, Select[IntegerPartitions[n], Length[#]>1&]}]];
DeleteCases[Table[Length[Select[qurt[n], SameQ[##, k]&@@Length/@Position[#, {}]&]], {n, 10}, {k, 0, n-1}], 0, {2}]

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); v[1]=x; [Vecrev(p/x) | p<-v]}
{ my(A=T(15)); for(n=1, #A, print(A[n])) } \\ Andrew Howroyd, Dec 09 2020

Crossrefs

Row sums are A120803. Second column is A083751. A regular version is A320179.

Cf. A000669, A005804, A048816, A119262, A120803, A141268, A244925, A319312.

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

Sequence in context: A146907 A322865 A238694 * A236939 A236936 A236915

Adjacent sequences: A320218 A320219 A320220 * A320222 A320223 A320224

Keyword

nonn,tabf

Author

Gus Wiseman, Oct 07 2018

Extensions

Terms a(36) and beyond from Andrew Howroyd, Dec 09 2020

Name clarified by Andrew Howroyd, Dec 09 2020

Status

approved