A290353 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the k-th Euler transform of the sequence with g.f. 1+x.

1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 2, 1, 0, 1, 1, 3, 3, 1, 0, 1, 1, 4, 6, 5, 1, 0, 1, 1, 5, 10, 14, 7, 1, 0, 1, 1, 6, 15, 30, 27, 11, 1, 0, 1, 1, 7, 21, 55, 75, 58, 15, 1, 0, 1, 1, 8, 28, 91, 170, 206, 111, 22, 1, 0, 1, 1, 9, 36, 140, 336, 571, 518, 223, 30, 1, 0

Offset: 0

Author

Alois P. Heinz, Jul 28 2017

Keywords

• nonn
• tabl

Comments

A(n,k) is the number of unlabeled rooted trees with exactly n leaves, all in level k. A(3,3) = 6:

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Examples

			Square array A(n,k) begins:
  1, 1,  1,   1,    1,    1,     1,     1,      1, ...
  1, 1,  1,   1,    1,    1,     1,     1,      1, ...
  0, 1,  2,   3,    4,    5,     6,     7,      8, ...
  0, 1,  3,   6,   10,   15,    21,    28,     36, ...
  0, 1,  5,  14,   30,   55,    91,   140,    204, ...
  0, 1,  7,  27,   75,  170,   336,   602,   1002, ...
  0, 1, 11,  58,  206,  571,  1337,  2772,   5244, ...
  0, 1, 15, 111,  518, 1789,  5026, 12166,  26328, ...
  0, 1, 22, 223, 1344, 5727, 19193, 54046, 133476, ...
		

Links

• Alois P. Heinz, Antidiagonals n = 0..140, flattened
• B. A. Huberman and T. Hogg, Complexity and adaptation, Evolution, games and learning (Los Alamos, N.M., 1985). Phys. D 22 (1986), no. 1-3, 376-384.
• Index entries for sequences related to rooted trees

Crossrefs

Columns k=1-10 give: A000012, A000041, A001970, A007713, A007714, A290355, A290356, A290357, A290358, A290359.

Rows 0+1,2-10 give: A000012, A001477, A000217, A000330, A007715, A290360, A290361, A290362, A290363, A290364.

Main diagonal gives A290354.

Cf. A144150.

Programs

Maple

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

Mathematica

A[n_, k_]:=b[n, k]=If[n<2, 1, If[k==0, 0, Sum[Sum[A[d, k - 1]*d, {d, Divisors[j]}] A[n - j, k], {j, n}]/n]]; Table[A[n, d - n], {d, 0, 14}, {n, 0, d}]//Flatten (* Indranil Ghosh, Jul 30 2017, after Maple code *)

Formula

G.f. of column k=0: 1+x, of column k>0: Product_{j>0} 1/(1-x^j)^A(j,k-1).