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%I #16 Mar 31 2023 07:18:28
%S 1,1,2,1,3,10,1,4,45,152,1,5,136,4743,7736,1,6,325,59008,3801411,
%T 1375952,1,7,666,426425,345706336,23938685973,877901648,1,8,1225,
%U 2164680,11782824375,28256240134144,1215663478473627,2046320373120,1,9,2080
%N T(n,k) = number of n X n 0..k zero-diagonal arrays with corresponding row and column sums equal.
%C Table starts
%C .........1................1....................1................1............1
%C .........2................3....................4................5............6
%C ........10...............45..................136..............325..........666
%C .......152.............4743................59008...........426425......2164680
%C ......7736..........3801411............345706336......11782824375.213067487016
%C ...1375952......23938685973.......28256240134144.7093199984236625
%C .877901648.1215663478473627.33097994593655140864
%H R. H. Hardin, <a href="/A229417/b229417.txt">Table of n, a(n) for n = 1..43</a>
%F Empirical for row n:
%F n=1: a(n) = 1
%F n=2: a(n) = n + 1
%F n=3: a(n) = (1/2)*n^4 + 2*n^3 + (7/2)*n^2 + 3*n + 1
%F n=4: [polynomial of degree 9]
%F Row n is an Ehrhart polynomial of degree (n-1)^2 for the polytope of x(i,j), i,j = 1..n for j <> i, with 0 <= x(i,j) <= 1 and Sum_i x(i,j) = Sum_i x(j,i). - _Robert Israel_, Mar 30 2023
%F T(n,k) = A229870(n,k) / (k + 1)^n. - _Andrew Howroyd_, Mar 30 2023
%e Some solutions for n=4 k=4
%e ..0..0..2..0....0..1..0..4....0..0..1..3....0..1..1..4....0..1..1..0
%e ..1..0..2..1....2..0..4..0....1..0..2..3....4..0..2..3....0..0..1..2
%e ..1..2..0..4....2..4..0..2....2..3..0..1....1..4..0..1....0..0..0..4
%e ..0..2..3..0....1..1..4..0....1..3..3..0....1..4..3..0....2..2..2..0
%Y Columns 1..3 are A007080, A229415, A229416.
%Y Rows 3..6 are A037270(n+1), A229418, A229419, A229420.
%Y Cf. A229870.
%K nonn,tabl
%O 1,3
%A _R. H. Hardin_, Sep 22 2013
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