A258419 Number of partitions of the 5-dimensional hypercube resulting from a sequence of n bisections, each of which splits any part perpendicular to any of the axes, such that each axis is used at least once.

5040, 230400, 6792750, 165293700, 3624918660, 74699100720, 1479942440340, 28577108044800, 542482698531000, 10181610525525360, 189663357076785270, 3515970161266821510, 64985380300281057950, 1199146771516702098500, 22111945264260791498090

Offset: 5

Author

Alois P. Heinz, May 29 2015

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 5..750

Crossrefs

Column k=5 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-> T(n, 5):
seq(a(n), n=5..25);

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_] := Sum[A[n, k - i]*(-1)^i*Binomial[k, i], {i, 0, k}];
a[n_] := T[n, 5];
a /@ Range[5, 25] (* Jean-François Alcover, Dec 11 2020, after Alois P. Heinz *)