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A258425
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Total number of partitions of all hypercubes resulting from a sequence of n bisections, each of which splits any part perpendicular to any of the axes, such that each dimension is used at least once.
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2
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1, 1, 6, 64, 1020, 21854, 590248, 19268098, 738194780, 32481348812, 1614506203400, 89478362311442, 5471239864890436, 365900668319641264, 26569358218427144576, 2081825562568924254126, 175078869470374599592604, 15730138729512408087404292
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OFFSET
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0,3
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LINKS
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FORMULA
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a(n) ~ 2^(2*n-5/8) * n^(n-1) / (exp(n) * (log(2))^(n+1)). - Vaclav Kotesovec, May 30 2015
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EXAMPLE
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a(2) = 2 + 4 = 6:
In one dimension: [||-], [-||]
. .___. .___. .___. .___.
In two dimensions: |_| | | |_| |_|_| |___|
. |_|_| |_|_| |___| |_|_| .
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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:= proc(n, k) option remember;
add(A(n, k-i)*(-1)^i*binomial(k, i), i=0..k)
end:
a:= n-> add(T(n, k), k=0..n):
seq(a(n), n=0..20);
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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_] := T[n, k] = Sum[A[n, k-i]*(-1)^i*Binomial[k, i], {i, 0, k}]; a[n_] := Sum[T[n, k], {k, 0, n}]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Feb 07 2017, translated from Maple *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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