A237019 Number of partitions of the 4-dimensional hypercube resulting from a sequence of n bisections, each of which splits any part perpendicular to any of the axes.

1, 4, 32, 314, 3440, 40320, 494736, 6274900, 81606432, 1082351600, 14583873120, 199075231680, 2747135823040, 38260367077504, 537108342893696, 7592185149935327, 107968131964541240, 1543633250073656032, 22174725274316816504, 319906758044330938320

Offset

0,2

Links

Alois P. Heinz, Table of n, a(n) for n = 0..500

Yu Hin (Gary) Au, Fatemeh Bagherzadeh, Murray R. Bremner, Enumeration and Asymptotic Formulas for Rectangular Partitions of the Hypercube, arXiv:1903.00813 [math.CO], Mar 03 2019.

Formula

G.f. G satisfies: x = Sum_{i=0..4} (-1)^i*C(4,i)*(G*x)^(2^(4-i)).

a(n) ~ 1 / (4 * sqrt(Pi) * sqrt(1 - 9*s^2 + 28*s^6 - 30*s^14) * n^(3/2) * r^(n + 1/2)), where r = 0.064125256179778049525781860636169050731447267306296991777... and s = 0.1318780179022368311092675722371337905927964507241063792485... are real roots of the system of equations s + 6*s^4 + s^16 = r + 4*(s^2 + s^8) and 1 + 24*s^3 + 16*s^15 = 8*(s + 4*s^7). - Vaclav Kotesovec, Jun 11 2018

Mathematica

Rest[CoefficientList[InverseSeries[Series[x*(1 - 4*x + 6*x^3 - 4*x^7 + x^15), {x, 0, 20}], x], x]] (* Vaclav Kotesovec, Jun 11 2018 *)

Crossrefs

Column k=4 of A237018.

Sequence in context: A294592 A197715 A369026 * A260155 A047734 A220118

Adjacent sequences: A237016 A237017 A237018 * A237020 A237021 A237022

Keyword

nonn

Author

Alois P. Heinz, Feb 02 2014

Status

approved