A258426 Number of partitions of the n-dimensional hypercube resulting from a sequence of 2n bisections, each of which splits any part perpendicular to any of the axes, such that each axis is used at least once.

1, 2, 184, 64464, 51622600, 74699100720, 171052924578480, 569565504689176800, 2601107886874207253760, 15609810973119409265234400, 119149819949135773678717267200, 1127426871984268618976053945104000, 12953029027945569352833762868999449600

Offset: 0

Author

Alois P. Heinz, May 29 2015

Keywords

• nonn

Examples

			a(1) = 2 : [||-],  [-||].
		

Links

• Alois P. Heinz and Vaclav Kotesovec, Table of n, a(n) for n = 0..139 (terms 0..70 from Alois P. Heinz)

Crossrefs

Cf. A255982, A258399.

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-> T(2*n,n):
seq(a(n), n=0..15);

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_] := T[2*n, n]; Table[a[n], {n, 0, 15}] (* Jean-François Alcover, Dec 18 2016, after Alois P. Heinz *)

Formula

a(n) = A255982(2n,n).

a(n) ~ c * d^n * n!^2 / n^(5/2), where d = A256254 = 98.8248737517356857317..., c = 2^(3/8) * (-LambertW(-2*exp(-2)))^(1/8) / (8 * Pi^(3/2) * sqrt(1 + LambertW(-2*exp(-2)))) = 0.033762267258894908009578351704834892... . - Vaclav Kotesovec, May 31 2015, updated Sep 27 2023