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%I #5 Mar 31 2012 12:36:45
%S 1,1,3,1,5,7,1,7,13,19,1,9,19,61,51,1,11,25,151,221,141,1,13,31,313,
%T 631,1001,393,1,15,37,571,1401,4621,4145,1107,1,17,43,949,2651,15681,
%U 23857,18733,3139,1,19,49,1471,4501,42821,90609,164599,82381,8953,1,21,55,2161
%N T(n,k)=Number of arrays of n integers in -k..k with sum zero and equal numbers of elements greater than zero and less than zero
%C Table starts
%C ....1......1.......1........1.........1..........1..........1...........1
%C ....3......5.......7........9........11.........13.........15..........17
%C ....7.....13......19.......25........31.........37.........43..........49
%C ...19.....61.....151......313.......571........949.......1471........2161
%C ...51....221.....631.....1401......2651.......4501.......7071.......10481
%C ..141...1001....4621....15681.....42821......99961.....207621......394241
%C ..393...4145...23857....90609....263201.....637393....1355145.....2613857
%C .1107..18733..164599...909945...3688091...12004357...33222463....81196529
%C .3139..82381..948871..6105913..27050251...93039589..266948431...668734321
%C .8953.375745.6359617.57290209.343631641.1554288913.5714583505.17932764577
%H R. H. Hardin, <a href="/A201811/b201811.txt">Table of n, a(n) for n = 1..9999</a>
%F Empirical for rows:
%F T(1,k) = 1
%F T(2,k) = 2*k + 1
%F T(3,k) = 6*k + 1
%F T(4,k) = 4*k^3 + 14*k + 1
%F T(5,k) = 20*k^3 + 30*k + 1
%F T(6,k) = 11*k^5 + 65*k^3 + 64*k + 1
%F T(7,k) = 77*k^5 + 175*k^3 + 140*k + 1
%F T(8,k) = (302/9)*k^7 + (2912/9)*k^5 + (3878/9)*k^3 + 318*k + 1
%F T(9,k) = 302*k^7 + 1064*k^5 + 1022*k^3 + 750*k + 1
%F T(10,k) = (15619/144)*k^9 + (37465/24)*k^7 + (146209/48)*k^5 + (86705/36)*k^3 + 1828*k + 1
%F T(11,k) = (171809/144)*k^9 + (48785/8)*k^7 + (386155/48)*k^5 + (206635/36)*k^3 + 4576*k + 1
%e Some solutions for n=7 k=3
%e ..0...-1...-3....2...-1...-2....3....0....1...-2....0....3...-2...-3....2...-3
%e .-1....1....1...-2....1....2....3....2...-1...-3...-2....0...-2....3....2....3
%e ..2....0....0....3...-2...-2...-3....2....1....0....2...-3....0...-1...-1....2
%e ..0....1....2...-3....2...-1....2...-2....0....3...-1...-1....1....1...-2...-3
%e ..1...-3...-1....3...-2....0....0....2....1...-3....1...-3...-3....0....1...-1
%e ..0....3....2...-3....0....2...-2...-2...-1....2....1....3....3....2...-2....0
%e .-2...-1...-1....0....2....1...-3...-2...-1....3...-1....1....3...-2....0....2
%Y Column 1 is A002426
%Y Row 2 is A004273(n+1)
%Y Row 3 is A016921
%K nonn,tabl
%O 1,3
%A _R. H. Hardin_ Dec 05 2011