A300613 Number of partitions of the n-dimensional hypercube resulting from a sequence of n n-sections, each of which divides any part perpendicular to any of the axes.

1, 1, 8, 324, 35840, 7906250, 2931489792, 1645181968584, 1302784779485184, 1384565648740109550, 1902231808400000000000, 3281726715984295577534536, 6946466406905591840863420416, 17702487251379919853870809258728, 53467000591059566447137539120168960

Offset

0,3

Links

Alois P. Heinz, Table of n, a(n) for n = 0..25

Example

a(2) = 8:
  ._______.  ._______.  ._______.  ._______.
  | | |   |  |   | | |  |_______|  |       |
  | | |   |  |   | | |  |_______|  |_______|
  | | |   |  |   | | |  |       |  |_______|
  |_|_|___|  |___|_|_|  |_______|  |_______|
  ._______.  ._______.  ._______.  ._______.
  |   |   |  |   |   |  |   |   |  |       |
  |___|   |  |   |___|  |___|___|  |_______|
  |   |   |  |   |   |  |       |  |   |   |
  |___|___|  |___|___|  |_______|  |___|___|.

Maple

a:= proc(n) option remember; `if`(n<2, 1, coeff(series(
      RootOf(x*(-1)^n=add(binomial(n, i)*(G*x)^(n^(n-i))*
       (-1)^i, i=0..n), G), x, n^2-n+1), x, n^2-n))
    end:
seq(a(n), n=0..12);
# second Maple program:
b:= proc(n, k, t, d) option remember; `if`(t=0, 1, `if`(t=1,
      g(n-1, k, d), add(g(j, k, d)*b(n-j-1, k, t-1, d), j=0..n-2)))
    end:
g:= proc(n, k, d) option remember; `if`(n=0, 1,
      -add(binomial(k, j)*(-1)^j*b(n+1, k, d^j, d), j=1..k))
    end:
a:= n-> g(n^2-n, n$2):
seq(a(n), n=0..14);

Mathematica

b[n_, k_, t_, d_] := b[n, k, t, d] = If[t == 0, 1, If[t == 1, g[n - 1, k, d], Sum[g[j, k, d] b[n - j - 1, k, t - 1, d], {j, 0, n - 2}]]];
g[n_, k_, d_] := g[n, k, d] = If[n == 0, 1, -Sum[Binomial[k, j] (-1)^j b[n + 1, k, d^j, d], {j, 1, k}]];
a[n_] := g[n^2 - n, n, n];
Table[Print[n, " ", a[n]]; a[n], {n, 0, 14}] (* Jean-François Alcover, Dec 30 2018, from second Maple program *)

Crossrefs

Cf. A300474.

Sequence in context: A171248 A344094 A264053 * A221407 A204069 A226551

Adjacent sequences: A300610 A300611 A300612 * A300614 A300615 A300616

Keyword

nonn

Author

Alois P. Heinz, Dec 15 2018

Status

approved