A318754 Number T(n,k) of rooted trees with n nodes such that k equals the maximal number of subtrees extending from the same node and having the same number of nodes; triangle T(n,k), n>=1, 0<=k<=n-1, read by rows.

1, 0, 1, 0, 1, 1, 0, 2, 1, 1, 0, 3, 4, 1, 1, 0, 6, 9, 3, 1, 1, 0, 12, 22, 9, 3, 1, 1, 0, 25, 54, 23, 8, 3, 1, 1, 0, 51, 139, 60, 23, 8, 3, 1, 1, 0, 111, 346, 166, 61, 22, 8, 3, 1, 1, 0, 240, 892, 447, 167, 61, 22, 8, 3, 1, 1, 0, 533, 2290, 1219, 461, 168, 60, 22, 8, 3, 1, 1

Offset

1,8

Comments

T(n,k) is defined for n,k >= 0. The triangle contains only the terms with k < n. T(n,k) = 0 for k >= n.

Links

Alois P. Heinz, Rows n = 1..200, flattened

Formula

T(n,k) = A318753(n,k) - A318753(n,k-1) for k > 0, A(n,0) = A063524(n).

Example

Triangle T(n,k) begins:
  1;
  0,   1;
  0,   1,   1;
  0,   2,   1,   1;
  0,   3,   4,   1,  1;
  0,   6,   9,   3,  1,  1;
  0,  12,  22,   9,  3,  1, 1;
  0,  25,  54,  23,  8,  3, 1, 1;
  0,  51, 139,  60, 23,  8, 3, 1, 1;
  0, 111, 346, 166, 61, 22, 8, 3, 1, 1;

Maple

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

Mathematica

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

Crossrefs

Columns k=0-10 give: A063524, A032305 (for n>1), A318817, A318818, A318819, A318820, A318821, A318822, A318823, A318824, A318825.

Row sums give A000081.

T(2n+2,n+1) give A255705.

Cf. A318753.

Sequence in context: A051509 A124816 A238349 * A318758 A334192 A124790

Adjacent sequences: A318751 A318752 A318753 * A318755 A318756 A318757

Keyword

nonn,tabl

Author

Alois P. Heinz, Sep 02 2018

Status

approved