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A223680 T(n,k)=Number of nXk 0..1 arrays with rows and antidiagonals unimodal.

%I A223680 #6 Jul 23 2025 04:23:25
%S A223680 2,4,4,7,16,8,11,49,64,16,16,121,316,256,32,22,256,1118,2032,1024,64,
%T A223680 29,484,3177,9822,13045,4096,128,37,841,7745,35509,85663,83737,16384,
%U A223680 256,46,1369,16857,105995,384009,744272,537496,65536,512,56,2116,33615,275775
%N A223680 T(n,k)=Number of nXk 0..1 arrays with rows and antidiagonals unimodal.
%C A223680 Table starts
%C A223680 ....2.......4.........7.........11..........16...........22............29
%C A223680 ....4......16........49........121.........256..........484...........841
%C A223680 ....8......64.......316.......1118........3177.........7745.........16857
%C A223680 ...16.....256......2032.......9822.......35509.......105995........275775
%C A223680 ...32....1024.....13045......85663......384009......1363639.......4123210
%C A223680 ...64....4096.....83737.....744272.....4106403.....17068664......58944337
%C A223680 ..128...16384....537496....6458585....43632367....210660192.....821284360
%C A223680 ..256...65536...3450100...56030742...462307835...2577807779...11265254628
%C A223680 ..512..262144..22145617..486038270..4893189359..31402790284..152970187735
%C A223680 .1024.1048576.142149013.4215998078.51766786082.381690187059.2064772010660
%H A223680 R. H. Hardin, <a href="/A223680/b223680.txt">Table of n, a(n) for n = 1..480</a>
%F A223680 Empirical for column k:
%F A223680 k=1: a(n) = 2*a(n-1)
%F A223680 k=2: a(n) = 4*a(n-1)
%F A223680 k=3: a(n) = 7*a(n-1) -3*a(n-2) -5*a(n-3) +2*a(n-4)
%F A223680 k=4: [order 9]
%F A223680 k=5: [order 19]
%F A223680 k=6: [order 36]
%F A223680 k=7: [order 70]
%F A223680 Empirical for row n:
%F A223680 n=1: a(n) = (1/2)*n^2 + (1/2)*n + 1
%F A223680 n=2: a(n) = (1/4)*n^4 + (1/2)*n^3 + (5/4)*n^2 + 1*n + 1
%F A223680 n=3: a(n) = (23/360)*n^6 + (31/120)*n^5 + (17/9)*n^4 + (23/24)*n^3 + (917/360)*n^2 + (77/60)*n + 1
%F A223680 n=4: polynomial of degree 8
%F A223680 n=5: polynomial of degree 10 for n>2
%F A223680 n=6: polynomial of degree 12 for n>3
%F A223680 n=7: polynomial of degree 14 for n>4
%e A223680 Some solutions for n=3 k=4
%e A223680 ..0..0..0..0....0..0..1..1....1..1..0..0....0..0..0..1....0..0..0..0
%e A223680 ..1..0..0..0....0..1..1..0....0..1..1..0....1..1..0..0....1..0..0..0
%e A223680 ..0..0..0..0....0..0..0..1....0..0..0..1....0..0..0..0....1..1..0..0
%Y A223680 Column 1 is A000079
%Y A223680 Column 2 is A000302
%Y A223680 Column 3 is A188868
%Y A223680 Row 1 is A000124
%Y A223680 Row 2 is A086601
%K A223680 nonn,tabl
%O A223680 1,1
%A A223680 _R. H. Hardin_ Mar 25 2013