A048809 -id:A048809 - cp's OEIS frontend

A244925 Number T(n,k) of n-node unlabeled rooted trees with every leaf at height k; triangle T(n,k), n>=1, 0<=k<=n-1, read by rows.

1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 2, 1, 1, 0, 1, 2, 2, 1, 1, 0, 1, 4, 3, 2, 1, 1, 0, 1, 4, 5, 3, 2, 1, 1, 0, 1, 7, 7, 6, 3, 2, 1, 1, 0, 1, 8, 12, 8, 6, 3, 2, 1, 1, 0, 1, 12, 18, 15, 9, 6, 3, 2, 1, 1, 0, 1, 14, 27, 23, 16, 9, 6, 3, 2, 1, 1, 0, 1, 21, 42, 39, 26, 17, 9, 6, 3, 2, 1, 1

Offset: 1

Alois P. Heinz, Jul 08 2014

			The A048816(5) = 5 rooted trees with 5 nodes with every leaf at the same height sorted by height are:
  :    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  :
  :         :             :       :     :
  : ---1--- : -----2----- : --3-- : -4- :
Thus row 5 = [0, 1, 2, 1, 1].
Triangle T(n,k) begins:
  1;
  0, 1;
  0, 1,  1;
  0, 1,  1,  1;
  0, 1,  2,  1,  1;
  0, 1,  2,  2,  1,  1;
  0, 1,  4,  3,  2,  1, 1;
  0, 1,  4,  5,  3,  2, 1, 1;
  0, 1,  7,  7,  6,  3, 2, 1, 1;
  0, 1,  8, 12,  8,  6, 3, 2, 1, 1;
  0, 1, 12, 18, 15,  9, 6, 3, 2, 1, 1;
  0, 1, 14, 27, 23, 16, 9, 6, 3, 2, 1, 1;
  ...

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

Columns k=0-10 give: A000007(n-1), A000012 (for n>0), A002865(n-1) (for n>2), A048808, A048809, A048810, A048811, A048812, A048813, A048814, A048815.
T(2n+1,n) gives A074045.
Row sums give A048816.

with(numtheory):
T:= proc(n, k) option remember; `if`(n=1, 1, `if`(k=0, 0,
      add(add(`if`(d

T[n_, k_] := T[n, k] = If[n == 1, 1, If[k == 0, 0, Sum[ Sum[ If[dJean-François Alcover, Jan 28 2015, after Alois P. Heinz *)

A048808 Number of rooted trees with n nodes with every leaf at height 3.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 12, 18, 27, 42, 64, 96, 146, 219, 327, 491, 730, 1084, 1608, 2376, 3500, 5154, 7563, 11076, 16193, 23625, 34395, 50005, 72550, 105089, 151984, 219448, 316362, 455434, 654661, 939736, 1347137, 1928593, 2757449, 3937675

Offset: 4

Christian G. Bower, Apr 15 1999

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 4..10000 (terms 4..1000 from Alois P. Heinz)
N. J. A. Sloane, Transforms
Index entries for sequences related to rooted trees

Cf. A048809-A048816.

Column k=3 of A244925.

T[n_, k_] := T[n, k] = If[n == 1, 1, If[k == 0, 0, Sum[Sum[If[d < k, 0, T[d, k - 1]*d], {d, Divisors[j]}]*T[n - j, k], {j, 1, n - 1}]/(n - 1)]];
a[n_] := T[n, 3];
Table[a[n], {n, 4, 50}] (* Jean-François Alcover, May 11 2019, after Alois P. Heinz in A244925 *)

Euler transform of A002865 (with a(0)=0) shifted right.

A048810 Number of rooted trees with n nodes with every leaf at height 5.

Original entry on oeis.org

1, 1, 2, 3, 6, 9, 16, 26, 44, 73, 123, 203, 340, 563, 935, 1550, 2571, 4251, 7034, 11618, 19188, 31654, 52201, 85999, 141631, 233074, 383375, 630215, 1035508, 1700501, 2791309, 4579587, 7510280, 12310980, 20172075, 33039130, 54092556

Offset: 6

Christian G. Bower, Apr 15 1999

nonn

Alois P. Heinz, Table of n, a(n) for n = 6..1000
N. J. A. Sloane, Transforms
Index entries for sequences related to rooted trees

Cf. A048808-A048816.

Column k=5 of A244925.

Euler transform of A048809 shifted right.

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A244925 Number T(n,k) of n-node unlabeled rooted trees with every leaf at height k; triangle T(n,k), n>=1, 0<=k<=n-1, read by rows.

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A048808 Number of rooted trees with n nodes with every leaf at height 3.

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A048810 Number of rooted trees with n nodes with every leaf at height 5.

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