A258419 - OEIS

%I #9 Dec 11 2020 03:43:07

%S 5040,230400,6792750,165293700,3624918660,74699100720,1479942440340,

%T 28577108044800,542482698531000,10181610525525360,189663357076785270,

%U 3515970161266821510,64985380300281057950,1199146771516702098500,22111945264260791498090

%N 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 Alois P. Heinz, <a href="/A258419/b258419.txt">Table of n, a(n) for n = 5..750</a>

%p b:= proc(n, k, t) option remember; `if`(t=0, 1, `if`(t=1,

%p        A(n-1, k), add(A(j, k)*b(n-j-1, k, t-1), j=0..n-2)))

%p     end:

%p A:= proc(n, k) option remember; `if`(n=0, 1,

%p       -add(binomial(k, j)*(-1)^j*b(n+1, k, 2^j), j=1..k))

%p     end:

%p T:= proc(n, k) option remember;

%p       add(A(n, k-i)*(-1)^i*binomial(k, i), i=0..k)

%p     end:

%p a:= n-> T(n, 5):

%p seq(a(n), n=5..25);

%t 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 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 T[n_, k_] := Sum[A[n, k - i]*(-1)^i*Binomial[k, i], {i, 0, k}];

%t a[n_] := T[n, 5];

%t a /@ Range[5, 25] (* _Jean-François Alcover_, Dec 11 2020, after _Alois P. Heinz_ *)

%Y Column k=5 of A255982.

%K nonn

%O 5,1

%A _Alois P. Heinz_, May 29 2015