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A250729 T(n,k)=Number of (n+1)X(k+1) 0..1 arrays with nondecreasing x(i,j)+x(i,j-1) in the i direction and nondecreasing min(x(i,j),x(i-1,j)) in the j direction.

%I A250729 #8 Jul 23 2025 12:44:40
%S A250729 9,22,18,50,46,33,114,110,85,58,257,257,208,144,99,579,596,496,365,
%T A250729 230,166,1302,1376,1158,885,600,350,275,2927,3173,2699,2092,1500,942,
%U A250729 513,452,6578,7310,6257,4889,3605,2434,1418,728,739,14782,16838,14520,11377,8514
%N A250729 T(n,k)=Number of (n+1)X(k+1) 0..1 arrays with nondecreasing x(i,j)+x(i,j-1) in the i direction and nondecreasing min(x(i,j),x(i-1,j)) in the j direction.
%C A250729 Table starts
%C A250729 ....9...22...50...114...257...579...1302...2927....6578...14782...33216
%C A250729 ...18...46..110...257...596..1376...3173...7310...16838...38777...89300
%C A250729 ...33...85..208...496..1158..2699...6257..14520...33640...77999..180744
%C A250729 ...58..144..365...885..2092..4889..11377..26419...61330..142336..330417
%C A250729 ...99..230..600..1500..3605..8514..19887..46315..107565..249853..579962
%C A250729 ..166..350..942..2434..6016.14437..34069..79704..185684..431691.1002869
%C A250729 ..275..513.1418..3807..9728.23941..57397.135645..317769..741367.1725118
%C A250729 ..452..728.2065..5760.15297.38821..95231.228455..540546.1268605.2963321
%C A250729 ..739.1006.2918..8465.23407.61554.155263.380220..912438.2161980.5081193
%C A250729 .1204.1358.4022.12119.34943.95438.248537.623913.1525255.3661515.8684030
%H A250729 R. H. Hardin, <a href="/A250729/b250729.txt">Table of n, a(n) for n = 1..1860</a>
%F A250729 Empirical for column k:
%F A250729 k=1: a(n) = 3*a(n-1) -2*a(n-2) -a(n-3) +a(n-4)
%F A250729 k=2: a(n) = 4*a(n-1) -5*a(n-2) +5*a(n-4) -4*a(n-5) +a(n-6); also a polynomial of degree 4 plus a quasipolynomial of degree 0 with period 2
%F A250729 k=3: a(n) = 4*a(n-1) -5*a(n-2) +5*a(n-4) -4*a(n-5) +a(n-6); also a polynomial of degree 4 plus a quasipolynomial of degree 0 with period 2
%F A250729 k=4: a(n) = 5*a(n-1) -9*a(n-2) +5*a(n-3) +5*a(n-4) -9*a(n-5) +5*a(n-6) -a(n-7) for n>8; also a polynomial of degree 5 plus a quasipolynomial of degree 0 with period 2 for n>1
%F A250729 k=5: [order 8; also a polynomial of degree 6 plus a quasipolynomial of degree 0 with period 2] for n>10
%F A250729 k=6: [order 9; also a polynomial of degree 7 plus a quasipolynomial of degree 0 with period 2] for n>14
%F A250729 k=7: [order 10; also a polynomial of degree 8 plus a quasipolynomial of degree 0 with period 2] for n>17
%F A250729 Empirical for row n:
%F A250729 n=1: a(n) = 3*a(n-1) -a(n-2) -2*a(n-3) +a(n-4)
%F A250729 n=2: a(n) = 2*a(n-1) +2*a(n-2) -3*a(n-3) for n>4
%F A250729 n=3: a(n) = 4*a(n-1) -2*a(n-2) -9*a(n-3) +12*a(n-4) -2*a(n-5) -3*a(n-6) +a(n-7) for n>8
%F A250729 n=4: [order 7] for n>9
%F A250729 n=5: [order 9] for n>12
%F A250729 n=6: [order 11] for n>15
%F A250729 n=7: [order 14] for n>19
%e A250729 Some solutions for n=4 k=4
%e A250729 ..0..0..0..0..1....0..0..0..0..0....0..0..0..0..0....0..0..0..0..0
%e A250729 ..1..0..1..0..1....0..0..0..0..0....0..0..0..0..0....0..0..0..0..0
%e A250729 ..0..1..0..1..0....0..1..0..0..0....0..0..0..0..0....0..0..0..0..0
%e A250729 ..1..0..1..0..1....1..0..1..0..1....0..0..0..0..0....0..0..0..0..1
%e A250729 ..0..1..0..1..0....0..1..0..1..0....0..1..0..1..0....0..1..1..0..1
%Y A250729 Column 1 is A192760(n+2)
%K A250729 nonn,tabl
%O A250729 1,1
%A A250729 _R. H. Hardin_, Nov 27 2014