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%I #6 Dec 18 2015 18:17:43
%S 1,0,3,0,3,6,0,1,13,10,0,0,12,37,15,0,0,4,62,87,21,0,0,0,46,253,178,
%T 28,0,0,0,13,407,804,328,36,0,0,0,1,387,2168,2136,558,45,0,0,0,0,230,
%U 3727,8685,4958,892,55,0,0,0,0,88,4257,23496,28376,10376,1357,66,0,0,0,0,20
%N T(n,k)=Number of ways to arrange k nonattacking knights on the lower triangle of an n X n board
%C Table starts
%C ...1....0......0........0.........0..........0...........0............0
%C ...3....3......1........0.........0..........0...........0............0
%C ...6...13.....12........4.........0..........0...........0............0
%C ..10...37.....62.......46........13..........1...........0............0
%C ..15...87....253......407.......387........230..........88...........20
%C ..21..178....804.....2168......3727.......4257........3300.........1739
%C ..28..328...2136.....8685.....23496......44005.......58630........56795
%C ..36..558...4958....28376....111433.....312296......641678.......986879
%C ..45..892..10376....79611....429343....1693828.....5025711.....11453482
%C ..55.1357..20013...198334...1407755....7449231....30209361.....95826217
%C ..66.1983..36144...449336...4061432...27785786...147914590....625250362
%C ..78.2803..61846...942072..10561723...90732814...614046090...3340670026
%C ..91.3853.101163..1852096..25208796..265594944..2226985986..15164906168
%C .105.5172.159286..3449261..55996244..709634275..7218141771..60136468319
%C .120.6802.242748..6133944.117021864.1755164932.21280624486.212856667131
%C .136.8788.359634.10482661.232070481.4063548824.57867863073.683992163441
%H R. H. Hardin, <a href="/A194492/b194492.txt">Table of n, a(n) for n = 1..421</a>
%F Empirical: T(n,k) is a polynomial of degree 2k in n for n>3*k-5, with lead coefficient 1/(2^k*k!)
%F T(n,1) = (1/2)*n^2 + (1/2)*n
%F T(n,2) = (1/8)*n^4 + (1/4)*n^3 - (17/8)*n^2 + (31/4)*n - 8 for n>1
%F T(n,3) = (1/48)*n^6 + (1/16)*n^5 - (17/16)*n^4 + (133/48)*n^3 + (433/24)*n^2 - (743/6)*n + 218 for n>4
%F T(n,4) = (1/384)*n^8 + (1/96)*n^7 - (17/64)*n^6 + (5/12)*n^5 + (1571/128)*n^4 - (6653/96)*n^3 - (8641/96)*n^2 + (13951/8)*n - 4161 for n>7
%F T(n,5) = (1/3840)*n^10 + (1/768)*n^9 - (17/384)*n^8 + (3/128)*n^7 + (951/256)*n^6 - (68939/3840)*n^5 - (1433/12)*n^4 + (230047/192)*n^3 - (149581/240)*n^2 - (718267/30)*n + 72425 for n>10
%F T(n,6) = (1/46080)*n^12 + (1/7680)*n^11 - (17/3072)*n^10 - (13/4608)*n^9 + (6635/9216)*n^8 - (66667/23040)*n^7 - (2096789/46080)*n^6 + (1744105/4608)*n^5 + (10251671/11520)*n^4 - (107575289/5760)*n^3 + (46056797/1440)*n^2 + (984185/3)*n - 1221248 for n>13
%F T(n,7) = (1/645120)*n^14 + (1/92160)*n^13 - (17/30720)*n^12 - (79/92160)*n^11 + (1047/10240)*n^10 - (29857/92160)*n^9 - (946249/92160)*n^8 + (48326573/645120)*n^7 + (7473253/15360)*n^6 - (153662159/23040)*n^5 - (1018307/768)*n^4 + (40217689/144)*n^3 - (212485407/280)*n^2 - (316047321/70)*n + 20362649 for n>16
%e Some solutions for n=5 k=4
%e ..0..........0..........0..........0..........0..........1..........0
%e ..0.1........0.1........0.0........0.0........0.1........0.0........1.0
%e ..0.0.0......1.1.1......1.1.0......0.0.0......0.0.0......0.0.0......0.0.0
%e ..0.0.0.0....0.0.0.0....1.1.0.0....0.1.0.0....0.0.0.0....1.1.0.0....0.0.0.1
%e ..0.1.0.1.1..0.0.0.0.0..0.0.0.0.0..0.1.1.0.1..1.1.1.0.0..0.0.0.0.1..1.0.1.0.0
%Y Column 1 is A000217
%K nonn,tabl
%O 1,3
%A _R. H. Hardin_ Aug 26 2011
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