A255982 -id:A255982 - cp's OEIS frontend

A258421 Number of partitions of the 7-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.

2162160, 196756560, 10778727960, 463305056760, 17266750912320, 586609859314080, 18699578507549520, 569565504689176800, 16777853060738524020, 482011338862966969980, 13586929812483090607600, 377442353035435719228120, 10367784656620152180344310

Offset: 7

Alois P. Heinz, May 29 2015

nonn

Alois P. Heinz, Table of n, a(n) for n = 7..700

Column k=7 of A255982.

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-> T(n, 7):
seq(a(n), n=7..25);

A258422 Number of partitions of the 8-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.

Original entry on oeis.org

57657600, 6895848960, 485566099200, 26364414061440, 1224007231940640, 51216101151626880, 1991943704397427200, 73440737647137519120, 2601107886874207253760, 89332305977055996111040, 2995343867463073686769440, 98555316817167057069129600

Offset: 8

Alois P. Heinz, May 29 2015

nonn

Alois P. Heinz, Table of n, a(n) for n = 8..650

Column k=8 of A255982.

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-> T(n, 8):
seq(a(n), n=8..25);

A258423 Number of partitions of the 9-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.

Original entry on oeis.org

1764322560, 268497815040, 23638153069440, 1582270134681600, 89523597871058400, 4521537191138385600, 210558053896067770200, 9231136974969952417200, 386479930120038746283600, 15609810973119409265234400, 612788961533595085909010880, 23513250306172521375772885440

Offset: 9

Alois P. Heinz, May 29 2015

nonn

Alois P. Heinz, Table of n, a(n) for n = 9..650

Column k=9 of A255982.

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-> T(n, 9):
seq(a(n), n=9..25);

A258424 Number of partitions of the 10-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.

Original entry on oeis.org

60949324800, 11504185056000, 1238502000960000, 100203614366688000, 6786584967157027200, 406962991813415247000, 22343812436173975084800, 1147985274106305649476000, 56030531363859577353444000, 2626132408521540739815456000, 119149819949135773678717267200

Offset: 10

Alois P. Heinz, May 29 2015

nonn

Alois P. Heinz, Table of n, a(n) for n = 10..600

Column k=10 of A255982.

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-> T(n, 10):
seq(a(n), n=10..25);

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.

Original entry on oeis.org

1, 1, 6, 64, 1020, 21854, 590248, 19268098, 738194780, 32481348812, 1614506203400, 89478362311442, 5471239864890436, 365900668319641264, 26569358218427144576, 2081825562568924254126, 175078869470374599592604, 15730138729512408087404292

Offset: 0

Alois P. Heinz, May 29 2015

nonn

			a(2) = 2 + 4 = 6:
In one dimension:    [||-],  [-||]
.                    .___.   .___.   .___.   .___.
In two dimensions:   |_| |   | |_|   |_|_|   |___|
.                    |_|_|   |_|_|   |___|   |_|_| .

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

Row sums of A255982.

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);

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 *)

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

cp's OEIS Frontend

A258421 Number of partitions of the 7-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.

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A258422 Number of partitions of the 8-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.

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A258423 Number of partitions of the 9-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.

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A258424 Number of partitions of the 10-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.

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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.

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