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A255982 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.

%I A255982 #24 Dec 17 2020 15:00:19
%S A255982 1,0,1,0,2,4,0,5,29,30,0,14,184,486,336,0,42,1148,5880,9744,5040,0,
%T A255982 132,7228,64464,192984,230400,95040,0,429,46224,679195,3279060,
%U A255982 6792750,6308280,2162160,0,1430,300476,7043814,51622600,165293700,259518600,196756560,57657600
%N A255982 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 A255982 Alois P. Heinz, <a href="/A255982/b255982.txt">Rows n = 0..135, flattened</a>
%F A255982 T(n,k) = Sum_{i=0..k} (-1)^i * C(k,i) * A237018(n,k-i).
%e A255982 A(3,1) = 5:
%e A255982   [||-|---], [-|||---], [-|-|-|-], [---|||-], [---|-||].
%e A255982 .
%e A255982 A(2,2) = 4:
%e A255982   ._______.  ._______.  ._______.  ._______.
%e A255982   |   |   |  |   |   |  |   |   |  |       |
%e A255982   |___|   |  |   |___|  |___|___|  |_______|
%e A255982   |   |   |  |   |   |  |       |  |   |   |
%e A255982   |___|___|  |___|___|  |_______|  |___|___|.
%e A255982 .
%e A255982 Triangle T(n,k) begins:
%e A255982   1
%e A255982   0,   1;
%e A255982   0,   2,     4;
%e A255982   0,   5,    29,     30;
%e A255982   0,  14,   184,    486,     336;
%e A255982   0,  42,  1148,   5880,    9744,    5040;
%e A255982   0, 132,  7228,  64464,  192984,  230400,   95040;
%e A255982   0, 429, 46224, 679195, 3279060, 6792750, 6308280, 2162160;
%e A255982   ...
%p A255982 b:= proc(n, k, t) option remember; `if`(t=0, 1, `if`(t=1,
%p A255982        A(n-1, k), add(A(j, k)*b(n-j-1, k, t-1), j=0..n-2)))
%p A255982     end:
%p A255982 A:= proc(n, k) option remember; `if`(n=0, 1,
%p A255982       -add(binomial(k, j)*(-1)^j*b(n+1, k, 2^j), j=1..k))
%p A255982     end:
%p A255982 T:= (n, k)-> add(A(n, k-i)*(-1)^i*binomial(k, i), i=0..k):
%p A255982 seq(seq(T(n, k), k=0..n), n=0..10);
%t A255982 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 A255982 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 A255982 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 A255982 Columns k=0-10 give: A000007, A000108 (for n>0), A258416, A258417, A258418, A258419, A258420, A258421, A258422, A258423, A258424.
%Y A255982 Main diagonal gives A001761.
%Y A255982 Row sums give A258425.
%Y A255982 T(2n,n) give A258426.
%Y A255982 Cf. A237018, A256061, A258427.
%K A255982 nonn,tabl
%O A255982 0,5
%A A255982 _Alois P. Heinz_, Mar 13 2015