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A250812 T(n,k)=Number of (n+1)X(k+1) 0..2 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 A250812 #6 Jul 23 2025 12:46:39
%S A250812 36,100,129,225,379,432,441,873,1315,1389,784,1731,3081,4321,4356,
%T A250812 1296,3097,6171,10233,13735,13449,2025,5139,11116,20631,32745,42769,
%U A250812 41112,3025,8049,18537,37333,66291,102393,131455,124869,4356,12043,29145,62469
%N A250812 T(n,k)=Number of (n+1)X(k+1) 0..2 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 A250812 Table starts
%C A250812 ......36.....100.....225......441......784.....1296.....2025......3025
%C A250812 .....129.....379.....873.....1731.....3097.....5139.....8049.....12043
%C A250812 .....432....1315....3081.....6171....11116....18537....29145.....43741
%C A250812 ....1389....4321...10233....20631....37333....62469....98481....148123
%C A250812 ....4356...13735...32745....66291...120304...201741...318585....479845
%C A250812 ...13449...42769..102393...207831...377857...634509..1003089...1512163
%C A250812 ...41112..131455..315561...641571..1167796..1962717..3104985...4683421
%C A250812 ..124869..400681..963513..1961031..3572173..6007149..9507441..14345803
%C A250812 ..377676.1214695.2924265..5955891.10854424.18260061.28908345..43630165
%C A250812 .1139169.3669409.8840313.18013431.32839417.55258029.87498129.132077683
%H A250812 R. H. Hardin, <a href="/A250812/b250812.txt">Table of n, a(n) for n = 1..241</a>
%F A250812 Empirical: T(n,k) = (((5/12)*k^4 + (11/3)*k^3 + (157/12)*k^2 + (95/6)*k + 6)*3^n - ((1/2)*k^4 + (7/2)*k^3 + (23/2)*k^2 + (17/2)*k)*2^n + (1/4)*k^4 + 1*k^3 + (9/4)*k^2 - (1/2)*k)/2
%F A250812 Empirical for column k:
%F A250812 k=1: a(n) = 6*a(n-1) -11*a(n-2) +6*a(n-3); a(n) = (39*3^n-24*2^n+3)/2
%F A250812 k=2: a(n) = 6*a(n-1) -11*a(n-2) +6*a(n-3); a(n) = (126*3^n-99*2^n+20)/2
%F A250812 k=3: a(n) = 6*a(n-1) -11*a(n-2) +6*a(n-3); a(n) = (304*3^n-264*2^n+66)/2
%F A250812 k=4: a(n) = 6*a(n-1) -11*a(n-2) +6*a(n-3); a(n) = (620*3^n-570*2^n+162)/2
%F A250812 k=5: a(n) = 6*a(n-1) -11*a(n-2) +6*a(n-3); a(n) = (1131*3^n-1080*2^n+335)/2
%F A250812 k=6: a(n) = 6*a(n-1) -11*a(n-2) +6*a(n-3); a(n) = (1904*3^n-1869*2^n+618)/2
%F A250812 k=7: a(n) = 6*a(n-1) -11*a(n-2) +6*a(n-3); a(n) = (3016*3^n-3024*2^n+1050)/2
%F A250812 Empirical for row n:
%F A250812 n=1: a(n) = (1/4)*n^4 + (5/2)*n^3 + (37/4)*n^2 + 15*n + 9
%F A250812 n=2: a(n) = 1*n^4 + 10*n^3 + 37*n^2 + 54*n + 27
%F A250812 n=3: a(n) = (15/4)*n^4 + 36*n^3 + (527/4)*n^2 + (359/2)*n + 81
%F A250812 n=4: a(n) = 13*n^4 + 121*n^3 + 439*n^2 + 573*n + 243
%F A250812 n=5: a(n) = (171/4)*n^4 + 390*n^3 + (5627/4)*n^2 + (3575/2)*n + 729
%F A250812 n=6: a(n) = 136*n^4 + 1225*n^3 + 4402*n^2 + 5499*n + 2187
%F A250812 n=7: a(n) = (1695/4)*n^4 + 3786*n^3 + (54287/4)*n^2 + (33539/2)*n + 6561
%e A250812 Some solutions for n=4 k=4
%e A250812 ..1..1..0..0..0....1..0..0..0..0....1..1..1..1..0....0..0..0..0..0
%e A250812 ..0..0..1..1..1....0..0..0..0..0....0..0..0..0..0....2..2..2..2..2
%e A250812 ..1..1..2..2..2....2..2..2..2..2....2..2..2..2..2....2..2..2..2..2
%e A250812 ..0..0..1..1..1....1..1..1..1..2....2..2..2..2..2....1..1..1..1..2
%e A250812 ..0..0..1..2..2....0..1..1..1..2....0..0..1..1..2....1..1..1..1..2
%Y A250812 Row 1 is A000537(n+2)
%K A250812 nonn,tabl
%O A250812 1,1
%A A250812 _R. H. Hardin_, Nov 27 2014