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

0,19

Links

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

Formula

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

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

Example

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, ...

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 *)

Crossrefs

Columns k=0-10 give: A063524, A004111, A248869, A318850, A318851, A318852, A318853, A318854, A318855, A318856, A318857.

Main diagonal gives A000081.

Cf. A318753, A318758.

Sequence in context: A131555 A293209 A318753 * A263834 A263754 A328800

Adjacent sequences: A318754 A318755 A318756 * A318758 A318759 A318760

Keyword

nonn,tabl

Author

Alois P. Heinz, Sep 02 2018

Status

approved