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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.
2
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
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
KEYWORD
nonn
AUTHOR
Alois P. Heinz, May 29 2015
STATUS
approved