cp's OEIS Frontend

A258426 Number of partitions of the n-dimensional hypercube resulting from a sequence of 2n bisections, each of which splits any part perpendicular to any of the axes, such that each axis is used at least once.

%I A258426 #23 Sep 27 2023 16:17:55
%S A258426 1,2,184,64464,51622600,74699100720,171052924578480,
%T A258426 569565504689176800,2601107886874207253760,15609810973119409265234400,
%U A258426 119149819949135773678717267200,1127426871984268618976053945104000,12953029027945569352833762868999449600
%N A258426 Number of partitions of the n-dimensional hypercube resulting from a sequence of 2n bisections, each of which splits any part perpendicular to any of the axes, such that each axis is used at least once.
%H A258426 Alois P. Heinz and Vaclav Kotesovec, <a href="/A258426/b258426.txt">Table of n, a(n) for n = 0..139</a> (terms 0..70 from Alois P. Heinz)
%F A258426 a(n) = A255982(2n,n).
%F A258426 a(n) ~ c * d^n * n!^2 / n^(5/2), where d = A256254 = 98.8248737517356857317..., c = 2^(3/8) * (-LambertW(-2*exp(-2)))^(1/8) / (8 * Pi^(3/2) * sqrt(1 + LambertW(-2*exp(-2)))) = 0.033762267258894908009578351704834892... . - _Vaclav Kotesovec_, May 31 2015, updated Sep 27 2023
%e A258426 a(1) = 2 : [||-],  [-||].
%p A258426 b:= proc(n, k, t) option remember; `if`(t=0, 1, `if`(t=1,
%p A258426        A(n-1, k), add(A(j, k)*b(n-j-1, k, t-1), j=0..n-2)))
%p A258426     end:
%p A258426 A:= proc(n, k) option remember; `if`(n=0, 1,
%p A258426       -add(binomial(k, j)*(-1)^j*b(n+1, k, 2^j), j=1..k))
%p A258426     end:
%p A258426 T:= proc(n, k) option remember;
%p A258426       add(A(n, k-i)*(-1)^i*binomial(k, i), i=0..k)
%p A258426     end:
%p A258426 a:= n-> T(2*n,n):
%p A258426 seq(a(n), n=0..15);
%t A258426 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_] := T[2*n, n]; Table[a[n], {n, 0, 15}] (* _Jean-François Alcover_, Dec 18 2016, after _Alois P. Heinz_ *)
%Y A258426 Cf. A255982, A258399.
%K A258426 nonn
%O A258426 0,2
%A A258426 _Alois P. Heinz_, May 29 2015