A252876 - OEIS

%I #23 Feb 09 2019 11:14:18

%S 0,0,0,1,1,1,3,8,8,3,6,26,44,26,6,10,61,153,153,61,10,15,120,413,615,

%T 413,120,15,21,211,949,1953,1953,949,211,21,28,343,1948,5281,7313,

%U 5281,1948,343,28,36,526,3676,12686,23203,23203,12686,3676,526,36,45,771,6497,27805,64920,85801,64920,27805,6497,771,45

%N T(n,k) = Number of n X k nonnegative integer arrays with upper left 0 and lower right n+k-4 and value increasing by 0 or 1 with every step right or down.

%C Table starts

%C ..0...0.....1......3......6......10.......15........21........28.........36

%C ..0...1.....8.....26.....61.....120......211.......343.......526........771

%C ..1...8....44....153....413.....949.....1948......3676......6497......10894

%C ..3..26...153....615...1953....5281....12686.....27805.....56624.....108549

%C ..6..61...413...1953...7313...23203....64920....164399....383735.....836797

%C .10.120...949...5281..23203...85801...277585....806347...2142634....5281314

%C .15.211..1948..12686..64920..277585..1030330...3407823..10237249...28340232

%C .21.343..3676..27805.164399..806347..3407823..12742873..42993671..132872804

%C .28.526..6497..56624.383735.2142634.10237249..42993671.161937617..555632319

%C .36.771.10894.108549.836797.5281314.28340232.132872804.555632319.2105918045

%H Alois P. Heinz, <a href="/A252876/b252876.txt">Table of n, a(n) for n = 1..2850</a> (first 479 terms from R. H. Hardin)

%H 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)

%F Empirical for column k:

%F k=1: a(n) = (1/2)*n^2 - (3/2)*n + 1

%F k=2: a(n) = (1/24)*n^4 + (5/12)*n^3 - (13/24)*n^2 - (11/12)*n + 1,

%F k=3: [polynomial of degree 6]

%F k=4: [polynomial of degree 8]

%F k=5: [polynomial of degree 10]

%F k=6: [polynomial of degree 12]

%F k=7: [polynomial of degree 14]

%F Empirical: with "n+k-3" instead of "n+k-4" T(n,k) = binomial(n+k,k) - 2.

%e Some solutions for n=3 k=4

%e ..0..1..1..1....0..0..1..1....0..1..2..3....0..0..1..1....0..0..1..1

%e ..1..1..2..2....0..1..1..2....1..1..2..3....0..0..1..2....0..1..2..2

%e ..1..1..2..3....1..2..2..3....1..2..2..3....1..1..2..3....1..1..2..3

%Y Columns 1-7 give: A000217(n-2), A252870, A252871, A252872, A252873, A252874, A252875.

%Y Main diagonal is A252869.

%Y Cf. A252976, A252930, A323846.

%K nonn,tabl

%O 1,7

%A _R. H. Hardin_, Dec 24 2014