A258425 Total number of partitions of all hypercubes resulting from a sequence of n bisections, each of which splits any part perpendicular to any of the axes, such that each dimension is used at least once.

1, 1, 6, 64, 1020, 21854, 590248, 19268098, 738194780, 32481348812, 1614506203400, 89478362311442, 5471239864890436, 365900668319641264, 26569358218427144576, 2081825562568924254126, 175078869470374599592604, 15730138729512408087404292

Offset: 0

Author

Alois P. Heinz, May 29 2015

Keywords

• nonn

Examples

			a(2) = 2 + 4 = 6:
In one dimension:    [||-],  [-||]
.                    .___.   .___.   .___.   .___.
In two dimensions:   |_| |   | |_|   |_|_|   |___|
.                    |_|_|   |_|_|   |___|   |_|_| .
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..135

Crossrefs

Row sums of A255982.

Programs

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:= proc(n, k) option remember;
      add(A(n, k-i)*(-1)^i*binomial(k, i), i=0..k)
    end:
a:= n-> add(T(n,k), k=0..n):
seq(a(n), n=0..20);

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_] := T[n, k] = Sum[A[n, k-i]*(-1)^i*Binomial[k, i], {i, 0, k}]; a[n_] := Sum[T[n, k], {k, 0, n}]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Feb 07 2017, translated from Maple *)

Formula

a(n) = Sum_{k=0..n} A255982(n,k).

a(n) ~ 2^(2*n-5/8) * n^(n-1) / (exp(n) * (log(2))^(n+1)). - Vaclav Kotesovec, May 30 2015