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A258417 Number of partitions of the 3-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.

%I A258417 #12 Dec 11 2020 03:43:34
%S A258417 30,486,5880,64464,679195,7043814,72707844,751082244,7785793080,
%T A258417 81092511276,849060054420,8937364804760,94564644817767,
%U A258417 1005496779910572,10740560345206680,115218669255806304,1240869923563291014,13412271463669969704,145454088924589697192
%N A258417 Number of partitions of the 3-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.
%H A258417 Alois P. Heinz, <a href="/A258417/b258417.txt">Table of n, a(n) for n = 3..1000</a>
%H A258417 Vaclav Kotesovec, <a href="/A258417/a258417.txt">Recurrence (of order 11)</a>
%F A258417 a(n) ~ c * d^n / n^(3/2), where d = 11.6335027253872064795086728699206569842475549795979388187955249065144... is the root of the equation 16777216 - 150994944*d + 1716387840*d^3 + 2063339520*d^4 - 6994944*d^5 - 21019200*d^6 + 454313*d^7 = 0 and c = 0.6170954330535517584816422123448632671500498041324155957832713069267... . - _Vaclav Kotesovec_, Feb 20 2016
%p A258417 b:= proc(n, k, t) option remember; `if`(t=0, 1, `if`(t=1,
%p A258417        A(n-1, k), add(A(j, k)*b(n-j-1, k, t-1), j=0..n-2)))
%p A258417     end:
%p A258417 A:= proc(n, k) option remember; `if`(n=0, 1,
%p A258417       -add(binomial(k, j)*(-1)^j*b(n+1, k, 2^j), j=1..k))
%p A258417     end:
%p A258417 T:= proc(n, k) option remember;
%p A258417       add(A(n, k-i)*(-1)^i*binomial(k, i), i=0..k)
%p A258417     end:
%p A258417 a:= n-> T(n, 3):
%p A258417 seq(a(n), n=3..25);
%t A258417 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}]]];
%t A258417 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 A258417 T[n_, k_] := Sum[A[n, k - i]*(-1)^i*Binomial[k, i], {i, 0, k}];
%t A258417 a[n_] := T[n, 3];
%t A258417 a /@ Range[3, 25] (* _Jean-François Alcover_, Dec 11 2020, after _Alois P. Heinz_ *)
%Y A258417 Column k=3 of A255982.
%K A258417 nonn
%O A258417 3,1
%A A258417 _Alois P. Heinz_, May 29 2015