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

0,13

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)

Links

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

Example

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, ...

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 *)

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.

Sequence in context: A127840 A017837 A145153 * A188816 A168312 A076837

Adjacent sequences: A255514 A255515 A255516 * A255518 A255519 A255520

Keyword

nonn,tabl

Author

Alois P. Heinz, Feb 24 2015

Status

approved