A300474 - OEIS

%I #35 Dec 29 2018 07:24:03

%S 1,1,8,96,2240,80960,4021248,255704064,19878918144,1829788646400,

%T 194788537180160,23556611967336448,3191162612827078656,

%U 478807179615908462592,78833945248222913495040,14133035289273287214366720,2740751307013005651817267200

%N Number of partitions of the square resulting from a sequence of n n-sections, each of which divides any part perpendicular to any of the axes.

%H Alois P. Heinz, <a href="/A300474/b300474.txt">Table of n, a(n) for n = 0..50</a>

%e a(2) = 8:

%e   ._______.  ._______.  ._______.  ._______.

%e   | | |   |  |   | | |  |_______|  |       |

%e   | | |   |  |   | | |  |_______|  |_______|

%e   | | |   |  |   | | |  |       |  |_______|

%e   |_|_|___|  |___|_|_|  |_______|  |_______|

%e   ._______.  ._______.  ._______.  ._______.

%e   |   |   |  |   |   |  |   |   |  |       |

%e   |___|   |  |   |___|  |___|___|  |_______|

%e   |   |   |  |   |   |  |       |  |   |   |

%e   |___|___|  |___|___|  |_______|  |___|___|.

%e   .

%p a:= proc(n) option remember; `if`(n<2, 1, coeff(series(

%p       RootOf(G-x-2*G^n+G^(n^2), G), x, n^2-n+2), x, n^2-n+1))

%p     end:

%p seq(a(n), n=0..16);

%t a[0] = a[1] = 1; a[n_] := Module[{G}, G[_] = 0; Do[G[x_] = 2 G[x]^n - G[x]^n^2 + x + O[x]^(n^2 - n + 2) // Normal, {n^2 - n + 2}];

%t Coefficient[G[x], x, n^2 - n + 1]];

%t Table[a[n], {n, 0, 16}] (* _Jean-François Alcover_, Dec 29 2018, after _Alois P. Heinz_ *)

%Y Cf. A091144, A236339, A237026, A300613, A322543.

%K nonn

%O 0,3

%A _Alois P. Heinz_, Dec 15 2018