A318757 Number A(n,k) of rooted trees with n nodes such that no more than k isomorphic subtrees extend from the same node; 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, 52, 0, 0, 1, 1, 2, 4, 9, 20, 46, 102, 190, 113, 0, 0, 1, 1, 2, 4, 9, 20, 47, 110, 250, 459, 247, 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, A004111, A248869, A318850, A318851, A318852, A318853, A318854, A318855, A318856, A318857.

Main diagonal gives A000081.

Cf. A318753, A318758.

Programs

Maple

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

Mathematica

h[n_, m_, t_, k_] := h[n, m, t, k] = If[m == 0, Binomial[n + t, t], If[n == 0, 0, Sum[h[n - 1, m - j, t + 1, k], {j, 1, Min[k, m]}]]];
b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, Sum[b[n - i*j, i - 1, k]*h[A[i, k], j, 0, k], {j, 0, n/i}]]];
A[n_, k_] := If[n < 2, n, b[n - 1, n - 1, k]];
Table[A[n, d - n], {d, 0, 14}, {n, 0, d}] // Flatten (* Jean-François Alcover, May 11 2019, after Alois P. Heinz *)

Formula

A(n,k) = Sum_{j=0..k} A318758(n,j) for n > 0.

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