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

2,1

Links

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

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)

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 *)

Crossrefs

Column k=2 of A255982.

Sequence in context: A301445 A353818 A198794 * A079885 A364404 A121191

Adjacent sequences: A258413 A258414 A258415 * A258417 A258418 A258419

Keyword

nonn

Author

Alois P. Heinz, May 29 2015

Status

approved