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R. H. Hardin, <a href="/A214595/b214595.txt">Table of n, a(n) for n = 1..1475</a>

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T(n,k)=Number of nXnXn triangular 0..k arrays with every horizontal row having the same average value

2, 3, 2, 4, 5, 2, 5, 8, 23, 2, 6, 13, 62, 401, 2, 7, 18, 157, 1862, 20351, 2, 8, 25, 312, 10177, 187862, 2869211, 2, 9, 32, 601, 33352, 3330677, 63120962, 1127599139, 2, 10, 41, 986, 103651, 20608352, 5495329427, 71200442882, 1248252244661, 2, 11, 50, 1619

1,1

Table starts

.2.....3......4.......5........6.........7.........8..........9.........10

.2.....5......8......13.......18........25........32.........41.........50

.2....23.....62.....157......312.......601.......986.......1619.......2426

.2...401...1862...10177....33352....103651....250042.....589763....1199614

.2.20351.187862.3330677.20608352.121537201.493575042.1877543213.5767190924

Empirical for row n:

n=1: a(k)=2*a(k-1)-a(k-2)

n=2: a(k)=2*a(k-1)-2*a(k-3)+a(k-4)

n=3: (order 12 antisymmetric)

n=4: (order 32 symmetric)

n=5: (order 84 symmetric)

Some solutions for n=4 k=4

.....2........1........2........2........2........2........2........2

....3.1......0.2......2.2......3.1......2.2......1.3......4.0......4.0

...3.2.1....0.3.0....3.2.1....2.4.0....0.2.4....3.0.3....1.2.3....4.0.2

..2.2.3.1..2.1.0.1..1.2.4.1..4.2.2.0..1.4.3.0..4.0.2.2..3.2.3.0..4.0.4.0

Row 2 is A000982(n+1)

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R. H. Hardin Jul 22 2012

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