A318753 Number A(n,k) of rooted trees with n nodes such that no more than k subtrees extending from the same node have the same number of 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, 1, 1, 0, 0, 1, 1, 2, 2, 0, 0, 1, 1, 2, 3, 3, 0, 0, 1, 1, 2, 4, 7, 6, 0, 0, 1, 1, 2, 4, 8, 15, 12, 0, 0, 1, 1, 2, 4, 9, 18, 34, 25, 0, 0, 1, 1, 2, 4, 9, 19, 43, 79, 51, 0, 0, 1, 1, 2, 4, 9, 20, 46, 102, 190, 111, 0, 0, 1, 1, 2, 4, 9, 20, 47, 110, 250, 457, 240, 0

Offset: 0

Author

Alois P. Heinz, Sep 02 2018

Keywords

• nonn
• tabl

Examples

			Square array A(n,k) begins:
  0,  0,  0,   0,   0,   0,   0,   0,   0, ...
  1,  1,  1,   1,   1,   1,   1,   1,   1, ...
  0,  1,  1,   1,   1,   1,   1,   1,   1, ...
  0,  1,  2,   2,   2,   2,   2,   2,   2, ...
  0,  2,  3,   4,   4,   4,   4,   4,   4, ...
  0,  3,  7,   8,   9,   9,   9,   9,   9, ...
  0,  6, 15,  18,  19,  20,  20,  20,  20, ...
  0, 12, 34,  43,  46,  47,  48,  48,  48, ...
  0, 25, 79, 102, 110, 113, 114, 115, 115, ...
		

Links

• Alois P. Heinz, Antidiagonals n = 0..200, flattened

Crossrefs

Columns k=0-10 give: A063524, A032305, A213920, A318797, A318798, A318799, A318800, A318801, A318802, A318803, A318804.

Rows n=0-2 give: A000004, A000012, A057427.

Main diagonal gives A000081.

Cf. A318754.

Programs

Maple

g:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
      binomial(A(i, k)+j-1, j)*g(n-i*j, i-1, k), j=0..min(k, n/i))))
    end:
A:= (n, k)-> g(n-1$2, k):
seq(seq(A(n, d-n), n=0..d), d=0..14);

Mathematica

g[n_, i_, k_] := g[n, i, k] = If[n == 0, 1, If[i < 1, 0, Sum[Binomial[A[i, k] + j - 1, j]*g[n - i*j, i - 1, k], {j, 0, Min[k, n/i]}]]];
A[n_, k_] := g[n - 1, n - 1, k];
Table[A[n, d - n], {d, 0, 14}, {n, 0, d}] // Flatten (* Jean-François Alcover, May 27 2019, after Alois P. Heinz *)

Formula

A(n,k) = Sum_{j=0..k} A318754(n,j) for n > 0.

A(n,n+j) = A000081(n) for j >= -1.