A244535 -id:A244535 - cp's OEIS frontend

A244530 Number T(n,k) of ordered unlabeled rooted trees with n nodes such that the minimal outdegree of inner nodes equals k; triangle T(n,k), n>=1, 0<=k<=n-1, read by rows.

1, 0, 1, 0, 1, 1, 0, 4, 0, 1, 0, 11, 2, 0, 1, 0, 36, 5, 0, 0, 1, 0, 117, 11, 3, 0, 0, 1, 0, 393, 28, 7, 0, 0, 0, 1, 0, 1339, 78, 8, 4, 0, 0, 0, 1, 0, 4630, 201, 21, 9, 0, 0, 0, 0, 1, 0, 16193, 532, 55, 10, 5, 0, 0, 0, 0, 1, 0, 57201, 1441, 121, 11, 11, 0, 0, 0, 0, 0, 1

Offset: 1

Joerg Arndt and Alois P. Heinz, Jun 29 2014

T(1,0) = 1 by convention.

			T(5,1) = 11:
  o   o     o     o     o     o     o     o     o     o     o
  |   |     |     |    / \   / \   / \    |    /|\   /|\   /|\
  o   o     o     o   o   o o   o o   o   o   o o o o o o o o o
  |   |    / \   / \  |         | |   |  /|\  |       |       |
  o   o   o   o o   o o         o o   o o o o o       o       o
  |  / \  |         | |         |
  o o   o o         o o         o
  |
  o
Triangle T(n,k) begins:
  1;
  0,     1;
  0,     1,   1;
  0,     4,   0,  1;
  0,    11,   2,  0,  1;
  0,    36,   5,  0,  0, 1;
  0,   117,  11,  3,  0, 0, 1;
  0,   393,  28,  7,  0, 0, 0, 1;
  0,  1339,  78,  8,  4, 0, 0, 0, 1;
  0,  4630, 201, 21,  9, 0, 0, 0, 0, 1;
  0, 16193, 532, 55, 10, 5, 0, 0, 0, 0, 1;

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

Columns k=0-10 give: A063524, A106640(n-2), A244531, A244532, A244533, A244534, A244535, A244536, A244537, A244538, A244539.
Row sums give A000108(n-1).
Cf. A244454 (unordered unlabeled rooted trees).

b:= proc(n, t, k) option remember; `if`(n=0,
      `if`(t in [0, k], 1, 0), `if`(t>n, 0, add(b(j-1, k$2)*
       b(n-j, max(0, t-1), k), j=1..n)))
    end:
T:= (n, k)-> b(n-1, k$2) -`if`(n=1 and k=0, 0, b(n-1, k+1$2)):
seq(seq(T(n, k), k=0..n-1), n=1..14);

b[n_, t_, k_] := b[n, t, k] = If[n == 0, If[t == 0 || t == k, 1, 0], If[t>n, 0, Sum[b[j-1, k, k]*b[n-j, Max[0, t-1], k], {j, 1, n}]]]; T[n_, k_] := b[n-1, k, k] - If[n == 1 && k == 0, 0, b[n-1, k+1, k+1]]; Table[Table[T[n, k], {k, 0, n-1}], {n, 1, 14}] // Flatten (* Jean-François Alcover, Jan 13 2015, translated from Maple *)

A244460 Number of unlabeled rooted trees with n nodes such that the minimal outdegree of inner nodes equals 6.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 1, 2, 2, 2, 2, 2, 4, 7, 12, 16, 21, 25, 34, 47, 70, 103, 147, 201, 276, 377, 527, 743, 1057, 1486, 2088, 2911, 4073, 5704, 8027, 11290, 15897, 22340, 31411, 44159, 62165, 87516, 123296, 173642, 244636, 344684, 485976, 685362, 966971, 1364301

Offset: 7

Joerg Arndt and Alois P. Heinz, Jun 29 2014

nonn

Alois P. Heinz, Table of n, a(n) for n = 7..900

Column k=6 of A244454.

Cf. A244535.

b:= proc(n, i, t, k) option remember; `if`(n=0, `if`(t in [0, k],
      1, 0), `if`(i<1 or t>n, 0, add(binomial(b((i-1)$2, k$2)+j-1, j)*
      b(n-i*j, i-1, max(0,t-j), k), j=0..n/i)))
    end:
a:= n-> b(n-1$2, 6$2) -b(n-1$2, 7$2):
seq(a(n), n=7..60);

b[n_, i_, t_, k_] := b[n, i, t, k] = If[n == 0, If[t == 0 || t == k, 1, 0], If[i < 1, 0, Sum[Binomial[b[i - 1, i - 1, k, k] + j - 1, j]*b[n - i*j, i - 1, Max[0, t - j], k], {j, 0, n/i}]] // FullSimplify]; a[n_] := b[n - 1, n - 1, 6, 6] - b[n - 1, n - 1, 7, 7]; Table[a[n], {n, 7, 60}] (* Jean-François Alcover, Feb 06 2015, after Maple *)

cp's OEIS Frontend

A244530 Number T(n,k) of ordered unlabeled rooted trees with n nodes such that the minimal outdegree of inner nodes equals k; triangle T(n,k), n>=1, 0<=k<=n-1, read by rows.

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