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A252828 T(n,k)=Number of nXk nonnegative integer arrays with upper left 0 and every value within 3 of its city block distance from the upper left and every value increasing by 0 or 1 with every step right or down.

%I A252828 #6 Jul 23 2025 13:40:45
%S A252828 1,2,2,4,6,4,8,18,18,8,15,53,81,53,15,26,142,340,340,142,26,42,339,
%T A252828 1238,1920,1238,339,42,64,729,3891,9075,9075,3891,729,64,93,1437,
%U A252828 10761,36292,54376,36292,10761,1437,93,130,2638,26764,125892,271846,271846,125892
%N A252828 T(n,k)=Number of nXk nonnegative integer arrays with upper left 0 and every value within 3 of its city block distance from the upper left and every value increasing by 0 or 1 with every step right or down.
%C A252828 Table starts
%C A252828 ...1....2......4.......8.......15........26.........42..........64...........93
%C A252828 ...2....6.....18......53......142.......339........729........1437.........2638
%C A252828 ...4...18.....81.....340.....1238......3891......10761.......26764........60988
%C A252828 ...8...53....340....1920.....9075.....36292.....125892......387849......1082111
%C A252828 ..15..142...1238....9075....54376....271846....1165921.....4396009.....14863460
%C A252828 ..26..339...3891...36292...271846...1679072....8807722....40232545....163307844
%C A252828 ..42..729..10761..125892..1165921...8807722...55960651...306796310...1481748658
%C A252828 ..64.1437..26764..387849..4396009..40232545..306796310..2001017650..11403395172
%C A252828 ..93.2638..60988.1082111.14863460.163307844.1481748658.11403395172..76084625352
%C A252828 .130.4568.129236.2777103.45791493.598768118.6411737114.57777817522.448101581256
%H A252828 R. H. Hardin, <a href="/A252828/b252828.txt">Table of n, a(n) for n = 1..721</a>
%F A252828 Empirical for column k:
%F A252828 k=1: a(n) = (1/6)*n^3 - (1/2)*n^2 + (4/3)*n
%F A252828 k=2: [polynomial of degree 6]
%F A252828 k=3: [polynomial of degree 9]
%F A252828 k=4: [polynomial of degree 12]
%F A252828 k=5: [polynomial of degree 15]
%F A252828 k=6: [polynomial of degree 18]
%F A252828 k=7: [polynomial of degree 21]
%F A252828 Empirical for "within 1" instead of "within 3" is T(n,k)=binomial(n+k,k)-1
%e A252828 Some solutions for n=4 k=4
%e A252828 ..0..0..0..1....0..1..2..3....0..0..1..2....0..0..1..1....0..1..2..2
%e A252828 ..0..1..1..2....1..2..2..3....1..1..2..2....0..1..1..1....1..1..2..2
%e A252828 ..0..1..2..3....2..3..3..4....1..2..2..3....0..1..2..2....1..1..2..2
%e A252828 ..1..2..3..4....3..3..3..4....2..2..2..3....1..2..2..3....1..2..2..3
%Y A252828 Column 1 is A000125(n-1)
%K A252828 nonn,tabl
%O A252828 1,2
%A A252828 _R. H. Hardin_, Dec 22 2014