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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.

%I A258425 #14 Feb 07 2017 08:28:13
%S A258425 1,1,6,64,1020,21854,590248,19268098,738194780,32481348812,
%T A258425 1614506203400,89478362311442,5471239864890436,365900668319641264,
%U A258425 26569358218427144576,2081825562568924254126,175078869470374599592604,15730138729512408087404292
%N 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.
%H A258425 Alois P. Heinz, <a href="/A258425/b258425.txt">Table of n, a(n) for n = 0..135</a>
%F A258425 a(n) = Sum_{k=0..n} A255982(n,k).
%F A258425 a(n) ~ 2^(2*n-5/8) * n^(n-1) / (exp(n) * (log(2))^(n+1)). - _Vaclav Kotesovec_, May 30 2015
%e A258425 a(2) = 2 + 4 = 6:
%e A258425 In one dimension:    [||-],  [-||]
%e A258425 .                    .___.   .___.   .___.   .___.
%e A258425 In two dimensions:   |_| |   | |_|   |_|_|   |___|
%e A258425 .                    |_|_|   |_|_|   |___|   |_|_| .
%p A258425 b:= proc(n, k, t) option remember; `if`(t=0, 1, `if`(t=1,
%p A258425        A(n-1, k), add(A(j, k)*b(n-j-1, k, t-1), j=0..n-2)))
%p A258425     end:
%p A258425 A:= proc(n, k) option remember; `if`(n=0, 1,
%p A258425       -add(binomial(k, j)*(-1)^j*b(n+1, k, 2^j), j=1..k))
%p A258425     end:
%p A258425 T:= proc(n, k) option remember;
%p A258425       add(A(n, k-i)*(-1)^i*binomial(k, i), i=0..k)
%p A258425     end:
%p A258425 a:= n-> add(T(n,k), k=0..n):
%p A258425 seq(a(n), n=0..20);
%t A258425 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 *)
%Y A258425 Row sums of A255982.
%K A258425 nonn
%O A258425 0,3
%A A258425 _Alois P. Heinz_, May 29 2015