A292086 Number T(n,k) of (unlabeled) rooted trees with n leaf nodes and without unary nodes such that k is the maximum of 1 and the node outdegrees; triangle T(n,k), n>=1, 1<=k<=n, read by rows.

1, 0, 1, 0, 1, 1, 0, 2, 2, 1, 0, 3, 6, 2, 1, 0, 6, 17, 7, 2, 1, 0, 11, 47, 22, 7, 2, 1, 0, 23, 133, 72, 23, 7, 2, 1, 0, 46, 380, 230, 77, 23, 7, 2, 1, 0, 98, 1096, 751, 256, 78, 23, 7, 2, 1, 0, 207, 3186, 2442, 861, 261, 78, 23, 7, 2, 1, 0, 451, 9351, 8006, 2897, 887, 262, 78, 23, 7, 2, 1

Offset

1,8

Links

Alois P. Heinz, Rows n = 1..141, flattened

Index entries for sequences related to rooted trees

Formula

T(n,k) = A292085(n,k) - A292085(n,k-1) for k>2, T(n,1) = A292085(n,1).

Example

:   T(4,2) = 2        :   T(4,3) = 2      : T(4,4) = 1 :
:                     :                   :            :
:       o       o     :      o       o    :     o      :
:      / \     / \    :     / \     /|\   :   /( )\    :
:     o   N   o   o   :    o   N   o N N  :  N N N N   :
:    / \     ( ) ( )  :   /|\     ( )     :            :
:   o   N    N N N N  :  N N N    N N     :            :
:  ( )                :                   :            :
:  N N                :                   :            :
:                     :                   :            :
Triangle T(n,k) begins:
  1;
  0,  1;
  0,  1,   1;
  0,  2,   2,   1;
  0,  3,   6,   2,  1;
  0,  6,  17,   7,  2,  1;
  0, 11,  47,  22,  7,  2, 1;
  0, 23, 133,  72, 23,  7, 2, 1;
  0, 46, 380, 230, 77, 23, 7, 2, 1;
  ...

Maple

b:= proc(n, i, v, k) option remember; `if`(n=0,
      `if`(v=0, 1, 0), `if`(i<1 or v<1 or n<v, 0,
      `if`(v=n, 1, add(binomial(A(i, k)+j-1, j)*
       b(n-i*j, i-1, v-j, k), j=0..min(n/i, v)))))
    end:
A:= proc(n, k) option remember; `if`(n<2, n,
      add(b(n, n+1-j, j, k), j=2..min(n, k)))
    end:
T:= (n, k)-> A(n, k)-`if`(k=1, 0, A(n, k-1)):
seq(seq(T(n, k), k=1..n), n=1..15);

Mathematica

b[n_, i_, v_, k_] := b[n, i, v, k] = If[n == 0, If[v == 0, 1, 0], If[i < 1 || v < 1 || n < v, 0, If[v == n, 1, Sum[Binomial[A[i, k] + j - 1, j]*b[n - i*j, i - 1, v - j, k], {j, 0, Min[n/i, v]}]]]];
A[n_, k_] := A[n, k] = If[n < 2, n, Sum[b[n, n + 1 - j, j, k], {j, 2, Min[n, k]}]];
T[n_, k_] := A[n, k] - If[k == 1, 0, A[n, k - 1]];
Table[Table[T[n, k], {k, 1, n}], {n, 1, 15}] // Flatten (* Jean-François Alcover, Nov 07 2017, after Alois P. Heinz *)

Crossrefs

Columns k=1-10 give: A063524, A001190 (for n>1), A292229, A292230, A292231, A292232, A292233, A292234, A292235, A292236.

Row sums give A000669.

Limit of reversed rows gives A292087.

Cf. A244372, A288942, A292085.

Sequence in context: A107424 A155161 A185937 * A065177 A064044 A213980

Adjacent sequences: A292083 A292084 A292085 * A292087 A292088 A292089

Keyword

nonn,tabl

Author

Alois P. Heinz, Sep 08 2017

Status

approved