A255517 Number A(n,k) of rooted identity trees with n nodes and k-colored non-root nodes; square array A(n,k), n>=0, k>=0, read by antidiagonals.

0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 2, 1, 0, 0, 1, 3, 5, 2, 0, 0, 1, 4, 12, 18, 3, 0, 0, 1, 5, 22, 64, 66, 6, 0, 0, 1, 6, 35, 156, 363, 266, 12, 0, 0, 1, 7, 51, 310, 1193, 2214, 1111, 25, 0, 0, 1, 8, 70, 542, 2980, 9748, 14043, 4792, 52, 0, 0, 1, 9, 92, 868, 6273, 30526, 82916, 91857, 21124, 113, 0

Offset: 0

Author

Alois P. Heinz, Feb 24 2015

Keywords

• nonn
• tabl

Comments

From Vaclav Kotesovec, Feb 24 2015: (Start)

k Limit n->infinity A(n,k)^(1/n)

1 2.517540352632003890795354598463447277335981266803... = A246169

2 5.249032491228170579164952216184309265343086337648... = A246312

3 7.969494030514425004826375511986491746399264355846...

4 10.688492754969652458452048798468242930479212456958...

5 13.407087472537747579787047072702638639945914705837...

6 16.125529360448558670505097146631763969697822205298...

7 18.843901825822305757579605844910623225182677164912...

8 21.562238702430237066018783115405680041128676137631...

9 24.280555694806692616578932533497629224907619468796...

10 26.998860838916733933849490675388336975888308433826...

100 271.64425688361559470587959030374804709717287744789...

Conjecture: For big k the limit asymptotically approaches k*exp(1).

(End)

Examples

			A(3,2) = 5:
  o    o    o    o      o
  |    |    |    |     / \
  1    1    2    2    1   2
  |    |    |    |
  1    2    1    2
Square array A(n,k) begins:
  0,  0,   0,    0,    0,     0,     0, ...
  1,  1,   1,    1,    1,     1,     1, ...
  0,  1,   2,    3,    4,     5,     6, ...
  0,  1,   5,   12,   22,    35,    51, ...
  0,  2,  18,   64,  156,   310,   542, ...
  0,  3,  66,  363, 1193,  2980,  6273, ...
  0,  6, 266, 2214, 9748, 30526, 77262, ...
		

Links

• Alois P. Heinz, Antidiagonals n = 0..140, flattened

Crossrefs

Columns k=1-10 give: A004111, A005753, A052757, A052772, A052797, A255518, A255519, A255520, A255521, A255522.

Rows n=0-4 give: A000004, A000012, A001477, A000326, 2*A051662(k-1) for k>0.

Lower diagonal gives A255523.

Cf. A242249, A256068.

Programs

Maple

with(numtheory):
A:= proc(n, k) option remember; `if`(n<2, n, add(A(n-j, k)*add(
      k*A(d, k)*d*(-1)^(j/d+1), d=divisors(j)), j=1..n-1)/(n-1))
    end:
seq(seq(A(n, d-n), n=0..d), d=0..14);

Mathematica

A[n_, k_] := A[n, k] = If[n<2, n, Sum[A[n-j, k]*Sum[k*A[d, k]*d*(-1)^(j/d + 1), {d, Divisors[j]}], {j, 1, n-1}]/(n-1)]; Table[Table[A[n, d-n], {n, 0, d}], {d, 0, 14}] // Flatten (* Jean-François Alcover, Feb 22 2016, after Alois P. Heinz *)