%I #13 Jun 19 2020 04:01:58

%S 1,1,1,1,3,3,1,3,5,1,1,5,15,15,5,1,5,21,29,21,9,1,7,27,75,95,53,7,1,7,

%T 47,115,239,255,109,11,1,9,57,217,529,883,747,239,13,1,9,67,297,1041,

%U 2369,3285,2215,489,13,1,11,97,479,1785,5429,10799,12619,6649,1051,23,1,11

%N T(n,k)=Number of zero-sum -k..k arrays of n elements with first through third differences also in -k..k.

%H R. H. Hardin, <a href="/A202511/b202511.txt">Table of n, a(n) for n = 1..716</a>

%e Table starts

%e ..1....1.....1......1.......1.......1........1........1.........1.........1

%e ..1....3.....3......5.......5.......7........7........9.........9........11

%e ..3....5....15.....21......27......47.......57.......67........97.......111

%e ..1...15....29.....75.....115.....217......297......479.......609.......887

%e ..5...21....95....239.....529....1041.....1785.....3009......4655......6937

%e ..9...53...255....883....2369....5429....11015....20519.....35743.....58863

%e ..7..109...747...3285...10799...29295....68781...145389....282771....514931

%e .11..239..2215..12619...50519..161469...436919..1046251...2272569...4571935

%e .13..489..6649..48079..235731..890659..2782455..7550637..18334273..40766453

%e .13.1051.19967.185533.1110857.4953959.17852623.54870161.148884915.365789931

%e ...

%e Some solutions for n=8 k=7

%e .-1...-2...-5...-4...-6...-6...-1...-2...-4...-4...-3...-5...-3...-4...-1...-3

%e .-4...-4...-4...-3...-6...-4...-5...-5...-3...-5...-6...-4...-1...-3...-3...-1

%e .-2...-5...-1....0...-5...-3...-3...-1...-1...-2...-4...-2...-2...-1...-2....4

%e .-2...-5....3....4....1....0...-1....3...-1....3...-1....2...-3...-1....1....5

%e ..2....1....3....7....5....3....4....3...-3....6....3....3....1....3....2....1

%e ..5....6....2....4....5....6....6....1....0....3....5....3....4....4....1...-2

%e ..4....5....0...-1....5....3....2...-1....5...-1....6....4....5....2...-1...-2

%e .-2....4....2...-7....1....1...-2....2....7....0....0...-1...-1....0....3...-2

%Y Row 3 is A201874

%K nonn,tabl

%O 1,5

%A _R. H. Hardin_, Dec 20 2011