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A215788 T(n,k)=Number of permutations of 0..floor((n*k-1)/2) on even squares of an nXk array such that each row, column, diagonal and (downwards) antidiagonal of even squares is increasing.

%I A215788 #8 Jul 22 2025 23:24:38
%S A215788 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,1,1,1,1,2,5,2,1,1,1,1,5,12,10,4,
%T A215788 1,1,1,1,5,42,29,25,4,1,1,1,1,14,110,262,189,50,8,1,1,1,1,14,462,932,
%U A215788 2465,458,125,8,1,1,1,1,42,1274,11694,26451,15485,2988,250,16,1,1,1,1,42,6006
%N A215788 T(n,k)=Number of permutations of 0..floor((n*k-1)/2) on even squares of an nXk array such that each row, column, diagonal and (downwards) antidiagonal of even squares is increasing.
%C A215788 Table starts
%C A215788 .1.1.1..1....1......1........1..........1...........1...........1...........1
%C A215788 .1.1.1..1....2......2........5..........5..........14..........14..........42
%C A215788 .1.1.1..2....5.....12.......42........110.........462........1274........6006
%C A215788 .1.1.1..2...10.....29......262........932.......11694.......46988......727846
%C A215788 .1.1.1..4...25....189.....2465......26451......530429.....7027942...187205626
%C A215788 .1.1.1..4...50....458....15485.....234217....14296434...297246092.26970790176
%C A215788 .1.1.1..8..125...2988...146205....6812794...673507749.48337803306
%C A215788 .1.1.1..8..250...7241...918637...60485308.18255280444
%C A215788 .1.1.1.16..625..47241..8674386.1761748159
%C A215788 .1.1.1.16.1250.114482.54503318
%C A215788 .1.1.1.32.3125.746892
%C A215788 .1.1.1.32.6250
%H A215788 R. H. Hardin, <a href="/A215788/b215788.txt">Table of n, a(n) for n = 1..140</a>
%F A215788 Empirical for column k:
%F A215788 k=4: a(n) = 2*a(n-2)
%F A215788 k=5: a(n) = 5*a(n-2)
%F A215788 k=6: a(n) = 16*a(n-2) -3*a(n-4)
%F A215788 k=7: a(n) = 61*a(n-2) -99*a(n-4) -2*a(n-6)
%F A215788 k=8: a(n) = 272*a(n-2) -3439*a(n-4) -3336*a(n-6) +140*a(n-8)
%F A215788 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 A215788 Some solutions for n=7 k=4
%e A215788 ..0..x..1..x....0..x..1..x....0..x..1..x....0..x..1..x....0..x..1..x
%e A215788 ..x..2..x..3....x..2..x..4....x..2..x..4....x..2..x..3....x..2..x..3
%e A215788 ..4..x..5..x....3..x..5..x....3..x..5..x....4..x..5..x....4..x..5..x
%e A215788 ..x..6..x..8....x..6..x..8....x..6..x..8....x..6..x..7....x..6..x..7
%e A215788 ..7..x..9..x....7..x..9..x....7..x..9..x....8..x..9..x....8..x..9..x
%e A215788 ..x.10..x.12....x.10..x.12....x.10..x.11....x.10..x.12....x.10..x.11
%e A215788 .11..x.13..x...11..x.13..x...12..x.13..x...11..x.13..x...12..x.13..x
%Y A215788 Column 5 is A026383(n-1)
%Y A215788 Row 2 is A000108(floor((n-1)/2))
%Y A215788 Odd squares: A215870
%K A215788 nonn,tabl
%O A215788 1,17
%A A215788 _R. H. Hardin_ Aug 23 2012