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A258425 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. 2
1, 1, 6, 64, 1020, 21854, 590248, 19268098, 738194780, 32481348812, 1614506203400, 89478362311442, 5471239864890436, 365900668319641264, 26569358218427144576, 2081825562568924254126, 175078869470374599592604, 15730138729512408087404292 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..135

FORMULA

a(n) = Sum_{k=0..n} A255982(n,k).

a(n) ~ 2^(2*n-5/8) * n^(n-1) / (exp(n) * (log(2))^(n+1)). - Vaclav Kotesovec, May 30 2015

EXAMPLE

a(2) = 2 + 4 = 6:

In one dimension:    [||-],  [-||]

.                    .___.   .___.   .___.   .___.

In two dimensions:   |_| |   | |_|   |_|_|   |___|

.                    |_|_|   |_|_|   |___|   |_|_| .

MAPLE

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);

MATHEMATICA

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-Fran├žois Alcover, Feb 07 2017, translated from Maple *)

CROSSREFS

Row sums of A255982.

Sequence in context: A173500 A141008 A336114 * A249592 A333983 A087488

Adjacent sequences:  A258422 A258423 A258424 * A258426 A258427 A258428

KEYWORD

nonn

AUTHOR

Alois P. Heinz, May 29 2015

STATUS

approved

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Last modified October 19 09:22 EDT 2021. Contains 348074 sequences. (Running on oeis4.)