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%I #26 Dec 18 2014 17:57:09
%S 2,3,3,4,6,4,5,10,10,5,6,15,20,15,6,7,21,35,35,21,7,8,28,56,70,56,28,
%T 8,9,36,84,126,126,84,36,9,10,45,120,210,252,210,120,45,10,11,55,165,
%U 330,462,462,330,165,55,11,12,66,220,495,792,924,792,495,220,66,12,13,78,286,715
%N T(n,k) = number of n X k binary matrices with rows in lexicographically nondecreasing order and columns in lexicographically nonincreasing order.
%C Table starts:
%C ..2..3...4....5....6....7.....8.....9....10.....11.....12.....13......14
%C ..3..6..10...15...21...28....36....45....55.....66.....78.....91.....105
%C ..4.10..20...35...56...84...120...165...220....286....364....455.....560
%C ..5.15..35...70..126..210...330...495...715...1001...1365...1820....2380
%C ..6.21..56..126..252..462...792..1287..2002...3003...4368...6188....8568
%C ..7.28..84..210..462..924..1716..3003..5005...8008..12376..18564...27132
%C ..8.36.120..330..792.1716..3432..6435.11440..19448..31824..50388...77520
%C ..9.45.165..495.1287.3003..6435.12870.24310..43758..75582.125970..203490
%C .10.55.220..715.2002.5005.11440.24310.48620..92378.167960.293930..497420
%C .11.66.286.1001.3003.8008.19448.43758.92378.184756.352716.646646.1144066
%C Is this (apart from offsets and formatting) the same sequence as A014410? [_R. J. Mathar_, Oct 02 2010]
%C Yes, because it obeys the recursion formula for binomial coefficients: the top left element is either 0 (leaving T(n-1,k) ways to fill the rest) or 1 (leaving T(n,k-1) ways to fill the rest). [_Karl W. Heuer_, Aug 25 2014]
%H R. H. Hardin, <a href="/A180986/b180986.txt">Table of n, a(n) for n=1..544</a>
%e All solutions for 3 X 3:
%e ..0..0..0....0..0..0....0..0..0....0..0..0....0..0..0....0..0..0....0..0..0
%e ..0..0..0....0..0..0....0..0..0....1..0..0....1..0..0....1..1..0....1..0..0
%e ..1..0..0....1..1..0....1..1..1....1..0..0....1..1..0....1..1..0....1..1..1
%e ...
%e ..0..0..0....0..0..0....1..0..0....1..0..0....1..0..0....1..0..0....1..0..0
%e ..1..1..0....1..1..1....1..0..0....1..0..0....1..1..0....1..0..0....1..1..0
%e ..1..1..1....1..1..1....1..0..0....1..1..0....1..1..0....1..1..1....1..1..1
%e ...
%e ..1..0..0....1..1..0....1..1..0....1..1..0....1..1..1....0..0..0
%e ..1..1..1....1..1..0....1..1..0....1..1..1....1..1..1....0..0..0
%e ..1..1..1....1..1..0....1..1..1....1..1..1....1..1..1....0..0..0
%Y A014410 is the same sequence viewed as a triangle.
%K nonn,tabl
%O 1,1
%A _R. H. Hardin_, Sep 30 2010
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