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

0,5

Links

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

Formula

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

Example

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;
  ...

Maple

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);

Mathematica

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 *)

Crossrefs

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.

Sequence in context: A319275 A269011 A274086 * A256061 A323099 A002652

Adjacent sequences: A255979 A255980 A255981 * A255983 A255984 A255985

Keyword

nonn,tabl

Author

Alois P. Heinz, Mar 13 2015

Status

approved