%I #4 Jul 29 2012 08:16:55

%S 2,3,2,4,4,2,5,6,6,2,6,9,14,14,2,7,12,29,50,38,2,8,16,50,182,242,146,

%T 2,9,20,88,458,1802,1682,578,2,10,25,136,1184,7550,29162,13442,2882,2,

%U 11,30,209,2490,31412,210914,657722,134402,14402,2,12,36,302,5213,100350

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

%C Table starts

%C .2..3...4....5....6.....7......8......9.....10......11......12.......13

%C .2..4...6....9...12....16.....20.....25.....30......36......42.......49

%C .2..6..14...29...50....88....136....209....302.....430.....584......793

%C .2.14..50..182..458..1184...2490...5213...9722...17864...30284....51088

%C .2.38.242.1802.7550.31412.100350.310079.811472.2065406.4695974.10458806

%H R. H. Hardin, <a href="/A214906/b214906.txt">Table of n, a(n) for n = 1..1004</a>

%F Empirical for row n:

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

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

%F n=3: a(k)=2*a(k-2)+2*a(k-3)-4*a(k-5)-3*a(k-6)+3*a(k-8)+4*a(k-9)-2*a(k-11)-2*a(k-12)+a(k-14)

%F n=4: (order 62 symmetric)

%e Some solutions for n=4 k=4

%e .....3........2........2........3........2........2........2........2

%e ....2.4......2.2......1.3......2.4......1.3......2.2......1.3......1.3

%e ...2.3.4....0.3.3....0.3.3....1.4.4....1.1.4....2.2.2....0.3.3....1.2.3

%e ..2.3.3.4..1.2.2.3..0.2.3.3..3.3.3.3..0.0.4.4..1.1.2.4..2.2.2.2..0.2.2.4

%Y Column 2 is A010551(n+1)+2

%Y Row 2 is A002620(n+2)

%K nonn,tabl

%O 1,1

%A _R. H. Hardin_ Jul 29 2012