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%I #31 Dec 30 2018 03:45:29
%S 1,1,8,324,35840,7906250,2931489792,1645181968584,1302784779485184,
%T 1384565648740109550,1902231808400000000000,3281726715984295577534536,
%U 6946466406905591840863420416,17702487251379919853870809258728,53467000591059566447137539120168960
%N 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.
%H Alois P. Heinz, <a href="/A300613/b300613.txt">Table of n, a(n) for n = 0..25</a>
%e a(2) = 8:
%e ._______. ._______. ._______. ._______.
%e | | | | | | | | |_______| | |
%e | | | | | | | | |_______| |_______|
%e | | | | | | | | | | |_______|
%e |_|_|___| |___|_|_| |_______| |_______|
%e ._______. ._______. ._______. ._______.
%e | | | | | | | | | | |
%e |___| | | |___| |___|___| |_______|
%e | | | | | | | | | | |
%e |___|___| |___|___| |_______| |___|___|.
%p a:= proc(n) option remember; `if`(n<2, 1, coeff(series(
%p RootOf(x*(-1)^n=add(binomial(n, i)*(G*x)^(n^(n-i))*
%p (-1)^i, i=0..n), G), x, n^2-n+1), x, n^2-n))
%p end:
%p seq(a(n), n=0..12);
%p # second Maple program:
%p b:= proc(n, k, t, d) option remember; `if`(t=0, 1, `if`(t=1,
%p g(n-1, k, d), add(g(j, k, d)*b(n-j-1, k, t-1, d), j=0..n-2)))
%p end:
%p g:= proc(n, k, d) option remember; `if`(n=0, 1,
%p -add(binomial(k, j)*(-1)^j*b(n+1, k, d^j, d), j=1..k))
%p end:
%p a:= n-> g(n^2-n, n$2):
%p seq(a(n), n=0..14);
%t 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}]]];
%t 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}]];
%t a[n_] := g[n^2 - n, n, n];
%t Table[Print[n, " ", a[n]]; a[n], {n, 0, 14}] (* _Jean-François Alcover_, Dec 30 2018, from second Maple program *)
%Y Cf. A300474.
%K nonn
%O 0,3
%A _Alois P. Heinz_, Dec 15 2018
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