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T(n,k)=Number of (n+1)X(k+1) 0..1 arrays x(i,j) with row sums sum{j*x(i,j), j=1..k+1} nondecreasing, and column sums sum{i*x(i,j), i=1..n+1} nondecreasing
7

%I #4 Nov 30 2013 17:37:07

%S 7,14,14,23,48,23,36,128,128,36,54,316,578,316,54,76,760,2329,2329,

%T 760,76,104,1736,8898,16684,8898,1736,104,138,3843,32279,111870,

%U 111870,32279,3843,138,179,8263,111609,706823,1366233,706823,111609,8263,179,227

%N T(n,k)=Number of (n+1)X(k+1) 0..1 arrays x(i,j) with row sums sum{j*x(i,j), j=1..k+1} nondecreasing, and column sums sum{i*x(i,j), i=1..n+1} nondecreasing

%C Table starts

%C ...7....14......23........36.........54.........76........104..........138

%C ..14....48.....128.......316........760.......1736.......3843.........8263

%C ..23...128.....578......2329.......8898......32279.....111609.......367415

%C ..36...316....2329.....16684.....111870.....706823....4251357.....24298903

%C ..54...760....8898....111870....1366233...15664761..170427953...1758001316

%C ..76..1736...32279....706823...15664761..330749381.6633342932.126048354885

%C .104..3843..111609...4251357..170427953.6633342932

%C .138..8263..367415..24298903.1758001316

%C .179.17414.1156446.132106886

%C .227.36175.3507097

%H R. H. Hardin, <a href="/A232831/b232831.txt">Table of n, a(n) for n = 1..84</a>

%e Some solutions for n=4 k=4

%e ..0..1..0..1..0....1..0..0..0..0....0..0..0..0..1....1..1..0..0..0

%e ..1..0..0..0..1....0..0..1..0..0....1..0..0..0..1....1..1..0..0..0

%e ..1..0..1..1..1....0..0..0..0..1....0..1..0..0..1....0..1..0..1..1

%e ..0..1..1..1..1....1..0..0..1..1....0..0..1..0..1....0..0..1..1..1

%e ..0..1..1..1..1....0..1..1..1..1....0..0..0..1..1....0..0..1..1..1

%K nonn,tabl

%O 1,1

%A _R. H. Hardin_, Nov 30 2013