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A250656 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 A250656 #6 Jul 23 2025 12:39:34
%S A250656 9,16,19,25,34,39,36,53,70,79,49,76,109,142,159,64,103,156,221,286,
%T A250656 319,81,134,211,316,445,574,639,100,169,274,427,636,893,1150,1279,121,
%U A250656 208,345,554,859,1276,1789,2302,2559,144,251,424,697,1114,1723,2556,3581
%N A250656 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 A250656 Table starts
%C A250656 ....9...16....25....36....49....64....81...100...121...144...169....196....225
%C A250656 ...19...34....53....76...103...134...169...208...251...298...349....404....463
%C A250656 ...39...70...109...156...211...274...345...424...511...606...709....820....939
%C A250656 ...79..142...221...316...427...554...697...856..1031..1222..1429...1652...1891
%C A250656 ..159..286...445...636...859..1114..1401..1720..2071..2454..2869...3316...3795
%C A250656 ..319..574...893..1276..1723..2234..2809..3448..4151..4918..5749...6644...7603
%C A250656 ..639.1150..1789..2556..3451..4474..5625..6904..8311..9846.11509..13300..15219
%C A250656 .1279.2302..3581..5116..6907..8954.11257.13816.16631.19702.23029..26612..30451
%C A250656 .2559.4606..7165.10236.13819.17914.22521.27640.33271.39414.46069..53236..60915
%C A250656 .5119.9214.14333.20476.27643.35834.45049.55288.66551.78838.92149.106484.121843
%H A250656 R. H. Hardin, <a href="/A250656/b250656.txt">Table of n, a(n) for n = 1..880</a>
%F A250656 Empirical: T(n,k) = 2^(n-1)*k^2 + (5*2^(n-1)-1)*k + 2^(n+1)
%F A250656 Empirical for column k:
%F A250656 k=1: a(n) = 3*a(n-1) -2*a(n-2); also a(n) = 2^(n-1) +(5*2^(n-1) -1) +2^(n+1)
%F A250656 k=2: a(n) = 3*a(n-1) -2*a(n-2); also a(n) = 2^(n-1)*4 +(5*2^(n-1) -1)*2 +2^(n+1)
%F A250656 k=3: a(n) = 3*a(n-1) -2*a(n-2); also a(n) = 2^(n-1)*9 +(5*2^(n-1) -1)*3 +2^(n+1)
%F A250656 k=4: a(n) = 3*a(n-1) -2*a(n-2); also a(n) = 2^(n-1)*16 +(5*2^(n-1) -1)*4 +2^(n+1)
%F A250656 k=5: a(n) = 3*a(n-1) -2*a(n-2); also a(n) = 2^(n-1)*25 +(5*2^(n-1) -1)*5 +2^(n+1)
%F A250656 k=6: a(n) = 3*a(n-1) -2*a(n-2); also a(n) = 2^(n-1)*36 +(5*2^(n-1) -1)*6 +2^(n+1)
%F A250656 k=7: a(n) = 3*a(n-1) -2*a(n-2); also a(n) = 2^(n-1)*49 +(5*2^(n-1) -1)*7 +2^(n+1)
%F A250656 Empirical for row n:
%F A250656 n=1: a(n) = 1*n^2 + 4*n + 4
%F A250656 n=2: a(n) = 2*n^2 + 9*n + 8
%F A250656 n=3: a(n) = 4*n^2 + 19*n + 16
%F A250656 n=4: a(n) = 8*n^2 + 39*n + 32
%F A250656 n=5: a(n) = 16*n^2 + 79*n + 64
%F A250656 n=6: a(n) = 32*n^2 + 159*n + 128
%F A250656 n=7: a(n) = 64*n^2 + 319*n + 256
%e A250656 Some solutions for n=4 k=4
%e A250656 ..1..1..0..1..1....0..0..0..0..0....0..0..0..0..0....1..1..1..0..0
%e A250656 ..0..0..0..1..1....1..1..1..1..1....1..1..1..1..1....0..0..0..0..0
%e A250656 ..0..0..0..1..1....1..1..1..1..1....0..0..0..0..0....0..0..0..0..0
%e A250656 ..0..0..0..1..1....0..0..0..0..0....1..1..1..1..1....1..1..1..1..1
%e A250656 ..0..0..0..1..1....0..1..1..1..1....1..1..1..1..1....0..0..0..1..1
%Y A250656 Column 1 is A052549(n+1)
%Y A250656 Column 2 is A176449
%Y A250656 Column 3 is A156127(n+1)
%Y A250656 Column 4 is A048487(n+2)
%Y A250656 Row 1 is A000290(n+2)
%Y A250656 Row 2 is A168244(n+3)
%K A250656 nonn,tabl
%O A250656 1,1
%A A250656 _R. H. Hardin_, Nov 26 2014