A258416 Number of partitions of the 2-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.

4, 29, 184, 1148, 7228, 46224, 300476, 1983102, 13266032, 89795420, 614058228, 4236652416, 29457698192, 206215486597, 1452248529432, 10281676045348, 73137772914324, 522472109334560, 3746685545297640, 26961148855455180, 194626321451800800, 1409026233004925340

Offset: 2

Author

Alois P. Heinz, May 29 2015

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 2..1000

Crossrefs

Column k=2 of A255982.

Programs

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-> T(n, 2):
seq(a(n), n=2..25);

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_] := Sum[A[n, k - i]*(-1)^i*Binomial[k, i], {i, 0, k}];
a[n_] := T[n, 2];
a /@ Range[2, 25] (* Jean-François Alcover, Dec 11 2020, after Alois P. Heinz *)

Formula

From Vaclav Kotesovec, May 29 2015: (Start)

Recurrence: 5*(n-2)*(n-1)*n*(n+1)*(13616*n^4 - 138092*n^3 + 514558*n^2 - 835288*n + 498441)*a(n) = - 6*(n-2)*(n-1)*n*(27232*n^5 - 289800*n^4 + 1170888*n^3 - 2195854*n^2 + 1802270*n - 411881)*a(n-1) + 16*(n-2)*(n-1)*(544640*n^6 - 6612960*n^5 + 32102192*n^4 - 79406652*n^3 + 104891690*n^2 - 69498516*n + 17766135)*a(n-2) - 8*(n-2)*(2*n - 5)*(1524992*n^6 - 18516288*n^5 + 89869136*n^4 - 222469596*n^3 + 295082666*n^2 - 197989116*n + 52268391)*a(n-3) - 16*(2*n - 7)*(2*n - 5)*(4*n - 13)*(4*n - 11)*(13616*n^4 - 83628*n^3 + 181978*n^2 - 165984*n + 53235)*a(n-4).

a(n) ~ c * d^n / (sqrt(Pi) * n^(3/2)), where d = 7.721133226857077553917531558... is the root of the equation 256 + 512*d - 32*d^2 - 5*d^3 = 0, c = 1.11097484883257916279675191289... is the root of the equation -8 + 364*c^2 - 518*c^4 + 185*c^6 = 0.

(End)