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%I #13 Nov 27 2015 05:33:57
%S 1,1,1,1,1,1,1,1,1,1,1,2,1,1,1,1,2,2,1,1,1,1,5,4,4,1,1,1,1,5,12,10,4,
%T 1,1,1,1,14,29,78,20,8,1,1,1,1,14,110,262,189,50,8,1,1,1,1,42,290,
%U 3001,1642,1233,100,16,1,1,1,1,42,1274,11694,26451,15485,2988,250,16,1,1,1,1,132
%N T(n,k) = Number of permutations of 0..floor((n*k-2)/2) on odd squares of an n X k array such that each row, column, diagonal and (downwards) antidiagonal of odd squares is increasing.
%C Table starts
%C .1.1.1..1....1......1.......1.........1.........1..........1........1
%C .1.1.1..2....2......5.......5........14........14.........42.......42
%C .1.1.1..2....4.....12......29.......110.......290.......1274.....3532
%C .1.1.1..4...10.....78.....262......3001.....11694.....170594...727846
%C .1.1.1..4...20....189....1642.....26451....307874....7027942.98057806
%C .1.1.1..8...50...1233...15485....767560..14296434.1124811332
%C .1.1.1..8..100...2988...97289...6812794.386699176
%C .1.1.1.16..250..19494..918637.198409297
%C .1.1.1.16..500..47241.5772013
%C .1.1.1.32.1250.308205
%C .1.1.1.32.2500
%C .1.1.1.64
%H R. H. Hardin, <a href="/A215870/b215870.txt">Table of n, a(n) for n = 1..125</a>
%F Empirical for column k:
%F k=4: a(n) = 2*a(n-2), A016116.
%F k=5: a(n) = 5*a(n-2) for n>3, A026395.
%F k=6: a(n) = 16*a(n-2) -3*a(n-4), A215866.
%F k=7: a(n) = 61*a(n-2) -99*a(n-4) -2*a(n-6), A215867.
%F k=8: a(n) = 272*a(n-2) -3439*a(n-4) -3336*a(n-6) +140*a(n-8).
%F k=9: a(n) = 1385*a(n-2) -131648*a(n-4) -318070*a(n-6) -4160916*a(n-8) -1097892*a(n-10) +648*a(n-12).
%e Some solutions for n=6, k=4:
%e ..x..0..x..1....x..0..x..2....x..0..x..2....x..0..x..1....x..0..x..1
%e ..2..x..3..x....1..x..3..x....1..x..3..x....2..x..3..x....2..x..3..x
%e ..x..4..x..5....x..4..x..6....x..4..x..5....x..4..x..6....x..4..x..6
%e ..6..x..7..x....5..x..7..x....6..x..7..x....5..x..7..x....5..x..7..x
%e ..x..8..x.10....x..8..x.10....x..8..x.10....x..8..x.10....x..8..x..9
%e ..9..x.11..x....9..x.11..x....9..x.11..x....9..x.11..x...10..x.11..x
%Y Column 5 is A026395(n-1).
%Y Row 2 is A000108(floor(n/2)).
%Y Even squares: A215788.
%K nonn,tabl
%O 1,12
%A _R. H. Hardin_, Aug 25 2012
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