A201811 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.

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, 631, 1001, 393, 1, 15, 37, 571, 1401, 4621, 4145, 1107, 1, 17, 43, 949, 2651, 15681, 23857, 18733, 3139, 1, 19, 49, 1471, 4501, 42821, 90609, 164599, 82381, 8953, 1, 21, 55, 2161

Offset: 1

Author

R. H. Hardin Dec 05 2011

Keywords

• nonn
• tabl

Comments

Table starts

....1......1.......1........1.........1..........1..........1...........1

....3......5.......7........9........11.........13.........15..........17

....7.....13......19.......25........31.........37.........43..........49

...19.....61.....151......313.......571........949.......1471........2161

...51....221.....631.....1401......2651.......4501.......7071.......10481

..141...1001....4621....15681.....42821......99961.....207621......394241

..393...4145...23857....90609....263201.....637393....1355145.....2613857

.1107..18733..164599...909945...3688091...12004357...33222463....81196529

.3139..82381..948871..6105913..27050251...93039589..266948431...668734321

.8953.375745.6359617.57290209.343631641.1554288913.5714583505.17932764577

Examples

			Some solutions for n=7 k=3
..0...-1...-3....2...-1...-2....3....0....1...-2....0....3...-2...-3....2...-3
.-1....1....1...-2....1....2....3....2...-1...-3...-2....0...-2....3....2....3
..2....0....0....3...-2...-2...-3....2....1....0....2...-3....0...-1...-1....2
..0....1....2...-3....2...-1....2...-2....0....3...-1...-1....1....1...-2...-3
..1...-3...-1....3...-2....0....0....2....1...-3....1...-3...-3....0....1...-1
..0....3....2...-3....0....2...-2...-2...-1....2....1....3....3....2...-2....0
.-2...-1...-1....0....2....1...-3...-2...-1....3...-1....1....3...-2....0....2
		

Links

• R. H. Hardin, Table of n, a(n) for n = 1..9999

Crossrefs

Column 1 is A002426

Row 2 is A004273(n+1)

Row 3 is A016921

Formula

Empirical for rows:

T(1,k) = 1

T(2,k) = 2*k + 1

T(3,k) = 6*k + 1

T(4,k) = 4*k^3 + 14*k + 1

T(5,k) = 20*k^3 + 30*k + 1

T(6,k) = 11*k^5 + 65*k^3 + 64*k + 1

T(7,k) = 77*k^5 + 175*k^3 + 140*k + 1

T(8,k) = (302/9)*k^7 + (2912/9)*k^5 + (3878/9)*k^3 + 318*k + 1

T(9,k) = 302*k^7 + 1064*k^5 + 1022*k^3 + 750*k + 1

T(10,k) = (15619/144)*k^9 + (37465/24)*k^7 + (146209/48)*k^5 + (86705/36)*k^3 + 1828*k + 1

T(11,k) = (171809/144)*k^9 + (48785/8)*k^7 + (386155/48)*k^5 + (206635/36)*k^3 + 4576*k + 1