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

0,2

Links

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

Example

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

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 *)

Crossrefs

Cf. A255982, A258399.

Sequence in context: A032593 A176477 A094483 * A163794 A232703 A135126

Adjacent sequences: A258423 A258424 A258425 * A258427 A258428 A258429

Keyword

nonn

Author

Alois P. Heinz, May 29 2015

Status

approved