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A255982
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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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16
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1, 0, 1, 0, 2, 4, 0, 5, 29, 30, 0, 14, 184, 486, 336, 0, 42, 1148, 5880, 9744, 5040, 0, 132, 7228, 64464, 192984, 230400, 95040, 0, 429, 46224, 679195, 3279060, 6792750, 6308280, 2162160, 0, 1430, 300476, 7043814, 51622600, 165293700, 259518600, 196756560, 57657600
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OFFSET
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0,5
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LINKS
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FORMULA
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T(n,k) = Sum_{i=0..k} (-1)^i * C(k,i) * A237018(n,k-i).
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EXAMPLE
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A(3,1) = 5:
[||-|---], [-|||---], [-|-|-|-], [---|||-], [---|-||].
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A(2,2) = 4:
._______. ._______. ._______. ._______.
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|___| | | |___| |___|___| |_______|
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|___|___| |___|___| |_______| |___|___|.
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Triangle T(n,k) begins:
1
0, 1;
0, 2, 4;
0, 5, 29, 30;
0, 14, 184, 486, 336;
0, 42, 1148, 5880, 9744, 5040;
0, 132, 7228, 64464, 192984, 230400, 95040;
0, 429, 46224, 679195, 3279060, 6792750, 6308280, 2162160;
...
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MAPLE
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b:= proc(n, k, t) option remember; `if`(t=0, 1, `if`(t=1,
A(n-1, k), add(A(j, k)*b(n-j-1, k, t-1), j=0..n-2)))
end:
A:= proc(n, k) option remember; `if`(n=0, 1,
-add(binomial(k, j)*(-1)^j*b(n+1, k, 2^j), j=1..k))
end:
T:= (n, k)-> add(A(n, k-i)*(-1)^i*binomial(k, i), i=0..k):
seq(seq(T(n, k), k=0..n), n=0..10);
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MATHEMATICA
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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_] := 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 *)
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CROSSREFS
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Columns k=0-10 give: A000007, A000108 (for n>0), A258416, A258417, A258418, A258419, A258420, A258421, A258422, A258423, A258424.
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KEYWORD
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AUTHOR
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STATUS
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approved
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