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%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
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