A258427 -id:A258427 - cp's OEIS frontend

A255982 Number T(n,k) of partitions of the k-dimensional hypercube resulting from a sequence of n bisections, each of which splits any part perpendicular to any of the axes, such that each axis is used at least once; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

1, 0, 1, 0, 2, 4, 0, 5, 29, 30, 0, 14, 184, 486, 336, 0, 42, 1148, 5880, 9744, 5040, 0, 132, 7228, 64464, 192984, 230400, 95040, 0, 429, 46224, 679195, 3279060, 6792750, 6308280, 2162160, 0, 1430, 300476, 7043814, 51622600, 165293700, 259518600, 196756560, 57657600

Offset: 0

Alois P. Heinz, Mar 13 2015

			A(3,1) = 5:
  [||-|---], [-|||---], [-|-|-|-], [---|||-], [---|-||].
.
A(2,2) = 4:
  ._______.  ._______.  ._______.  ._______.
  |   |   |  |   |   |  |   |   |  |       |
  |___|   |  |   |___|  |___|___|  |_______|
  |   |   |  |   |   |  |       |  |   |   |
  |___|___|  |___|___|  |_______|  |___|___|.
.
Triangle T(n,k) begins:
  1
  0,   1;
  0,   2,     4;
  0,   5,    29,     30;
  0,  14,   184,    486,     336;
  0,  42,  1148,   5880,    9744,    5040;
  0, 132,  7228,  64464,  192984,  230400,   95040;
  0, 429, 46224, 679195, 3279060, 6792750, 6308280, 2162160;
  ...

Alois P. Heinz, Rows n = 0..135, flattened

Columns k=0-10 give: A000007, A000108 (for n>0), A258416, A258417, A258418, A258419, A258420, A258421, A258422, A258423, A258424.
Main diagonal gives A001761.
Row sums give A258425.
T(2n,n) give A258426.
Cf. A237018, A256061, A258427.

b:= proc(n, k, t) option remember; `if`(t=0, 1, `if`(t=1,
       A(n-1, k), add(A(j, k)*b(n-j-1, k, t-1), j=0..n-2)))
    end:
A:= proc(n, k) option remember; `if`(n=0, 1,
      -add(binomial(k, j)*(-1)^j*b(n+1, k, 2^j), j=1..k))
    end:
T:= (n, k)-> add(A(n, k-i)*(-1)^i*binomial(k, i), i=0..k):
seq(seq(T(n, k), k=0..n), n=0..10);

b[n_, k_, t_] := b[n, k, t] = If[t == 0, 1, If[t == 1, A[n-1, k], Sum[ A[j, k]*b[n-j-1, k, t-1], {j, 0, n-2}]]];
A[n_, k_] := A[n, k] = If[n == 0, 1, -Sum[Binomial[k, j]*(-1)^j*b[n+1, k, 2^j], {j, 1, k}]];
T[n_, k_] := Sum[A[n, k-i]*(-1)^i*Binomial[k, i], {i, 0, k}]; Table[Table[T[n, k], {k, 0, n}], {n, 0, 10}] // Flatten (* Jean-François Alcover, Feb 20 2016, after Alois P. Heinz *)

T(n,k) = Sum_{i=0..k} (-1)^i * C(k,i) * A237018(n,k-i).

A256061 Number T(n,k) of 2n-length strings of balanced parentheses of exactly k different types; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 2, 4, 0, 5, 30, 30, 0, 14, 196, 504, 336, 0, 42, 1260, 6300, 10080, 5040, 0, 132, 8184, 71280, 205920, 237600, 95040, 0, 429, 54054, 774774, 3603600, 7207200, 6486480, 2162160, 0, 1430, 363220, 8288280, 58378320, 180180000, 273873600, 201801600, 57657600

Offset: 0

Alois P. Heinz, Mar 13 2015

Also number of binary trees with n inner nodes of exactly k different dimensions.  T(2,2) = 4:
: balanced parentheses :  ([]) :  [()] : ()[]  : []()  :
:----------------------:-------:-------:-------:-------:
:                trees :   (1) :   [2] : (1)   : [2]   :
:                      :   / \ :   / \ : / \   : / \   :
:                      : [2]   : (1)   :   [2] :   (1) :
:                      : / \   : / \   :   / \ :   / \ :

			A(3,2) = 30: (())[], (()[]), (([])), ()()[], ()([]), ()[()], ()[[]], ()[](), ()[][], ([()]), ([[]]), ([]()), ([])(), ([])[], ([][]), [(())], [()()], [()[]], [()](), [()][], [([])], [[()]], [[]()], [[]](), [](()), []()(), []()[], []([]), [][()], [][]().
Triangle T(n,k) begins:
  1;
  0,   1;
  0,   2,     4;
  0,   5,    30,     30;
  0,  14,   196,    504,     336;
  0,  42,  1260,   6300,   10080,    5040;
  0, 132,  8184,  71280,  205920,  237600,   95040;
  0, 429, 54054, 774774, 3603600, 7207200, 6486480, 2162160;
  ...

Alois P. Heinz, Rows n = 0..140, flattened

Columns k=0-1 give: A000007, A000108 (for n>0).
Main diagonal gives A001761.
Cf. A253180, A255982, A258427, A290605.

ctln:= proc(n) option remember; binomial(2*n, n)/(n+1) end:
A:= proc(n, k) option remember; k^n*ctln(n) end:
T:= (n, k)-> add(A(n, k-i)*(-1)^i*binomial(k, i), i=0..k):
seq(seq(T(n, k), k=0..n), n=0..10);

A[0, 0] = 1; A[n_, k_] := A[n, k] = k^n*CatalanNumber[n]; T[n_, k_] := Sum[A[n, k-i]*(-1)^i*Binomial[k, i], {i, 0, k}]; Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Feb 20 2017, translated from Maple *)

T(n,k) = Sum_{i=0..k} (-1)^i * C(k,i) * (k-i)^n * A000108(n).
T(n,k) = k! * A253180(n,k).
T(n,k) = Sum_{i=0..k} (-1)^i * C(k,i) * A290605(n,k-i). - Alois P. Heinz, Oct 28 2019

cp's OEIS Frontend

A255982 Number T(n,k) of partitions of the k-dimensional hypercube resulting from a sequence of n bisections, each of which splits any part perpendicular to any of the axes, such that each axis is used at least once; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

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