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

0,3

Links

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

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

Example

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

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 *)

Crossrefs

Row sums of A255982.

Sequence in context: A336114 A354494 A370894 * A365054 A249592 A333983

Adjacent sequences: A258422 A258423 A258424 * A258426 A258427 A258428

Keyword

nonn

Author

Alois P. Heinz, May 29 2015

Status

approved