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A252976 T(n,k) = Number of n X k nonnegative integer arrays with upper left 0 and lower right n+k-5 and value increasing by 0 or 1 with every step right or down.

%I A252976 #14 Nov 24 2015 05:20:11
%S A252976 0,0,0,0,0,0,1,1,1,1,4,13,18,13,4,10,61,153,153,61,10,20,192,770,1236,
%T A252976 770,192,20,35,483,2859,6997,6997,2859,483,35,56,1050,8694,30802,
%U A252976 46812,30802,8694,1050,56,84,2058,22924,112877,248182,248182,112877,22924,2058
%N A252976 T(n,k) = Number of n X k nonnegative integer arrays with upper left 0 and lower right n+k-5 and value increasing by 0 or 1 with every step right or down.
%C A252976 Table starts
%C A252976 ..0....0......0.......1........4........10.........20..........35...........56
%C A252976 ..0....0......1......13.......61.......192........483........1050.........2058
%C A252976 ..0....1.....18.....153......770......2859.......8694.......22924........54272
%C A252976 ..1...13....153....1236.....6997.....30802.....112877......359550......1024773
%C A252976 ..4...61....770....6997....46812....248182....1100210.....4230324.....14477724
%C A252976 .10..192...2859...30802...248182...1592348....8528422....39423196....161160206
%C A252976 .20..483...8694..112877..1100210...8528422...54926890...303382053...1471499970
%C A252976 .35.1050..22924..359550..4230324..39423196..303382053..1988261908..11360377192
%C A252976 .56.2058..54272.1024773.14477724.161160206.1471499970.11360377192..75922639116
%C A252976 .84.3732.118057.2667554.44951694.593478797.6383377435.57644900961.447545856560
%H A252976 R. H. Hardin, <a href="/A252976/b252976.txt">Table of n, a(n) for n = 1..449</a>
%H A252976 R. J. Mathar, <a href="http://vixra.org/abs/1511.0225">Counting 2-way monotonic terrace forms over rectangular landscapes</a>, vixra 1511.0225 (2015), subtable T_{n X m}(n+m-5).
%F A252976 Empirical for column k:
%F A252976 k=1: a(n) = (1/6)*n^3 - 1*n^2 + (11/6)*n - 1
%F A252976 k=2: [polynomial of degree 6]
%F A252976 k=3: [polynomial of degree 9]
%F A252976 k=4: [polynomial of degree 12]
%F A252976 k=5: [polynomial of degree 15]
%F A252976 k=6: [polynomial of degree 18]
%F A252976 k=7: [polynomial of degree 21]
%F A252976 Empirical: with "n+k-3" instead of "n+k-5" T(n,k) = binomial(n+k,k) - 2, see A166810, A166812, A166813.
%e A252976 Some solutions for n=4, k=4:
%e A252976 ..0..0..0..0....0..0..0..1....0..0..1..1....0..1..1..1....0..1..2..3
%e A252976 ..0..0..0..1....0..1..1..1....1..1..1..2....0..1..1..1....0..1..2..3
%e A252976 ..0..1..1..2....1..2..2..2....1..2..2..3....1..1..2..2....0..1..2..3
%e A252976 ..1..1..2..3....1..2..2..3....1..2..3..3....1..2..2..3....1..1..2..3
%Y A252976 Cf. A252876, A252930. Column 1 is A000292(n-3). Cf. A252970-A252975 (columns 2-7).
%K A252976 nonn,tabl
%O A252976 1,11
%A A252976 _R. H. Hardin_, Dec 25 2014