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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.

%I A258419 #9 Dec 11 2020 03:43:07
%S A258419 5040,230400,6792750,165293700,3624918660,74699100720,1479942440340,
%T A258419 28577108044800,542482698531000,10181610525525360,189663357076785270,
%U A258419 3515970161266821510,64985380300281057950,1199146771516702098500,22111945264260791498090
%N 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.
%H A258419 Alois P. Heinz, <a href="/A258419/b258419.txt">Table of n, a(n) for n = 5..750</a>
%p A258419 b:= proc(n, k, t) option remember; `if`(t=0, 1, `if`(t=1,
%p A258419        A(n-1, k), add(A(j, k)*b(n-j-1, k, t-1), j=0..n-2)))
%p A258419     end:
%p A258419 A:= proc(n, k) option remember; `if`(n=0, 1,
%p A258419       -add(binomial(k, j)*(-1)^j*b(n+1, k, 2^j), j=1..k))
%p A258419     end:
%p A258419 T:= proc(n, k) option remember;
%p A258419       add(A(n, k-i)*(-1)^i*binomial(k, i), i=0..k)
%p A258419     end:
%p A258419 a:= n-> T(n, 5):
%p A258419 seq(a(n), n=5..25);
%t A258419 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}]]];
%t A258419 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 A258419 T[n_, k_] := Sum[A[n, k - i]*(-1)^i*Binomial[k, i], {i, 0, k}];
%t A258419 a[n_] := T[n, 5];
%t A258419 a /@ Range[5, 25] (* _Jean-François Alcover_, Dec 11 2020, after _Alois P. Heinz_ *)
%Y A258419 Column k=5 of A255982.
%K A258419 nonn
%O A258419 5,1
%A A258419 _Alois P. Heinz_, May 29 2015