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Number T(n,k) of partitions of the k-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; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
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%I #24 Dec 17 2020 15:00:19

%S 1,0,1,0,2,4,0,5,29,30,0,14,184,486,336,0,42,1148,5880,9744,5040,0,

%T 132,7228,64464,192984,230400,95040,0,429,46224,679195,3279060,

%U 6792750,6308280,2162160,0,1430,300476,7043814,51622600,165293700,259518600,196756560,57657600

%N Number T(n,k) of partitions of the k-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; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

%H Alois P. Heinz, <a href="/A255982/b255982.txt">Rows n = 0..135, flattened</a>

%F T(n,k) = Sum_{i=0..k} (-1)^i * C(k,i) * A237018(n,k-i).

%e A(3,1) = 5:

%e [||-|---], [-|||---], [-|-|-|-], [---|||-], [---|-||].

%e .

%e A(2,2) = 4:

%e ._______. ._______. ._______. ._______.

%e | | | | | | | | | | |

%e |___| | | |___| |___|___| |_______|

%e | | | | | | | | | | |

%e |___|___| |___|___| |_______| |___|___|.

%e .

%e Triangle T(n,k) begins:

%e 1

%e 0, 1;

%e 0, 2, 4;

%e 0, 5, 29, 30;

%e 0, 14, 184, 486, 336;

%e 0, 42, 1148, 5880, 9744, 5040;

%e 0, 132, 7228, 64464, 192984, 230400, 95040;

%e 0, 429, 46224, 679195, 3279060, 6792750, 6308280, 2162160;

%e ...

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

%p seq(seq(T(n, k), k=0..n), n=0..10);

%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}]; Table[Table[T[n, k], {k, 0, n}], {n, 0, 10}] // Flatten (* _Jean-François Alcover_, Feb 20 2016, after _Alois P. Heinz_ *)

%Y Columns k=0-10 give: A000007, A000108 (for n>0), A258416, A258417, A258418, A258419, A258420, A258421, A258422, A258423, A258424.

%Y Main diagonal gives A001761.

%Y Row sums give A258425.

%Y T(2n,n) give A258426.

%Y Cf. A237018, A256061, A258427.

%K nonn,tabl

%O 0,5

%A _Alois P. Heinz_, Mar 13 2015