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A252820 T(n,k)=Number of nXk nonnegative integer arrays with upper left 0 and every value within 2 of its city block distance from the upper left and every value increasing by 0 or 1 with every step right or down.

%I A252820 #6 Jul 23 2025 13:40:18
%S A252820 1,2,2,4,6,4,7,17,17,7,11,40,63,40,11,16,81,187,187,81,16,22,147,468,
%T A252820 684,468,147,22,29,246,1032,2078,2078,1032,246,29,37,387,2067,5490,
%U A252820 7564,5490,2067,387,37,46,580,3840,13015,23664,23664,13015,3840,580,46,56,836,6716
%N A252820 T(n,k)=Number of nXk nonnegative integer arrays with upper left 0 and every value within 2 of its city block distance from the upper left and every value increasing by 0 or 1 with every step right or down.
%C A252820 Table starts
%C A252820 ..1...2.....4......7.....11......16.......22........29........37.........46
%C A252820 ..2...6....17.....40.....81.....147......246.......387.......580........836
%C A252820 ..4..17....63....187....468....1032.....2067......3840......6716......11179
%C A252820 ..7..40...187....684...2078....5490....13015.....28299.....57338.....109549
%C A252820 .11..81...468...2078...7564...23664....65711....165685....385736.....839799
%C A252820 .16.147..1032...5490..23664...86724...279300....809349...2147638....5289321
%C A252820 .22.246..2067..13015..65711..279300..1033761...3414257..10248688...28359679
%C A252820 .29.387..3840..28299.165685..809349..3414257..12755742..43017980..132916561
%C A252820 .37.580..6716..57338.385736.2147638.10248688..43017980.161986236..555724696
%C A252820 .46.836.11179.109549.839799.5289321.28359679.132916561.555724696.2106102800
%H A252820 R. H. Hardin, <a href="/A252820/b252820.txt">Table of n, a(n) for n = 1..1680</a>
%F A252820 Empirical for column k:
%F A252820 k=1: a(n) = (1/2)*n^2 - (1/2)*n + 1
%F A252820 k=2: a(n) = (1/24)*n^4 + (5/12)*n^3 - (1/24)*n^2 + (7/12)*n + 1
%F A252820 k=3: [polynomial of degree 6]
%F A252820 k=4: [polynomial of degree 8]
%F A252820 k=5: [polynomial of degree 10]
%F A252820 k=6: [polynomial of degree 12]
%F A252820 k=7: [polynomial of degree 14]
%F A252820 Empirical for "within 1" instead of "within 2" is T(n,k)=binomial(n+k,k)-1
%e A252820 Some solutions for n=4 k=4
%e A252820 ..0..1..2..2....0..1..2..3....0..0..1..1....0..1..2..3....0..0..0..1
%e A252820 ..1..1..2..3....1..1..2..3....0..1..2..2....1..2..3..3....0..1..1..2
%e A252820 ..2..2..3..4....1..1..2..3....1..2..3..3....2..3..3..4....1..2..2..3
%e A252820 ..2..3..4..5....1..2..3..4....1..2..3..4....2..3..4..4....2..2..3..4
%Y A252820 Column 1 is A000124(n-1)
%K A252820 nonn,tabl
%O A252820 1,2
%A A252820 _R. H. Hardin_, Dec 22 2014