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A250853 T(n,k)=Number of (n+1)X(k+1) 0..3 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 A250853 #6 Jul 23 2025 12:48:45
%S A250853 100,400,543,1225,2457,2670,3136,8037,13097,12311,7056,21436,44797,
%T A250853 63631,54410,14400,49599,123016,223933,291165,233683,27225,103293,
%U A250853 290646,626416,1043885,1280447,983950,48400,198297,614965,1499679,2955136
%N A250853 T(n,k)=Number of (n+1)X(k+1) 0..3 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 A250853 Table starts
%C A250853 ......100.......400.......1225.......3136........7056.......14400.......27225
%C A250853 ......543......2457.......8037......21436.......49599......103293......198297
%C A250853 .....2670.....13097......44797.....123016......290646......614965.....1195457
%C A250853 ....12311.....63631.....223933.....626416.....1499679.....3204951.....6279401
%C A250853 ....54410....291165....1043885....2955136.....7134786....15344785....30214465
%C A250853 ...233683...1280447....4648157...13263136....32201019....69543783...137379337
%C A250853 ...983950...5480917...20067117...57570016...140301126...303858745...601566177
%C A250853 ..4085631..23024631...84805533..244213216...596722599..1294875471..2567402601
%C A250853 .16796370..95448605..353060845.1019415136..2495502666..5422612945.10763029505
%C A250853 .68555723.391939087.1454214877.4206874336.10311967539.22429374423.44552408777
%H A250853 R. H. Hardin, <a href="/A250853/b250853.txt">Table of n, a(n) for n = 1..160</a>
%F A250853 Empirical T(n,k) = (((31/36)*k^6+(25/2)*k^5+(1229/18)*k^4+(620/3)*k^3+(10759/36)*k^2+(1181/6)*k+48)*4^n -((5/3)*k^6+(133/6)*k^5+(320/3)*k^4+(1717/6)*k^3+(944/3)*k^2+(344/3)*k)*3^n +(k^6+12*k^5+47*k^4+103*k^3+54*k^2-13*k)*2^n -((1/9)*k^6+(3/2)*k^5+(25/9)*k^4+(13/6)*k^3-(89/9)*k^2+(4/3)*k))/12
%F A250853 Empirical for column k:
%F A250853 k=1: a(n) = 10*a(n-1) -35*a(n-2) +50*a(n-3) -24*a(n-4); a(n) = (832*4^n-846*3^n+204*2^n+2)/12
%F A250853 k=2: a(n) = 10*a(n-1) -35*a(n-2) +50*a(n-3) -24*a(n-4); a(n) = (4838*4^n-6300*3^n+2214*2^n-80)/12
%F A250853 k=3: a(n) = 10*a(n-1) -35*a(n-2) +50*a(n-3) -24*a(n-4); a(n) = (18104*4^n-26144*3^n+10680*2^n-644)/12
%F A250853 k=4: a(n) = 10*a(n-1) -35*a(n-2) +50*a(n-3) -24*a(n-4); a(n) = (52650*4^n-80640*3^n+35820*2^n-2688)/12
%F A250853 k=5: a(n) = 10*a(n-1) -35*a(n-2) +50*a(n-3) -24*a(n-4); a(n) = (129528*4^n-206190*3^n+96660*2^n-8190)/12
%F A250853 k=6: a(n) = 10*a(n-1) -35*a(n-2) +50*a(n-3) -24*a(n-4); a(n) = (282492*4^n-462196*3^n+224994*2^n-20568)/12
%F A250853 k=7: a(n) = 10*a(n-1) -35*a(n-2) +50*a(n-3) -24*a(n-4); a(n) = (562288*4^n-939120*3^n+470064*2^n-45220)/12
%F A250853 Empirical for row n:
%F A250853 n=1: a(n) = (1/36)*n^6 + (1/2)*n^5 + (133/36)*n^4 + (43/3)*n^3 + (277/9)*n^2 + (104/3)*n + 16
%F A250853 n=2: a(n) = (2/9)*n^6 + (47/12)*n^5 + (953/36)*n^4 + (1141/12)*n^3 + (6527/36)*n^2 + 172*n + 64
%F A250853 n=3: a(n) = (3/2)*n^6 + (74/3)*n^5 + (621/4)*n^4 + (3161/6)*n^3 + (3691/4)*n^2 + 783*n + 256
%F A250853 n=4: a(n) = (76/9)*n^6 + (1595/12)*n^5 + (28765/36)*n^4 + (31373/12)*n^3 + (155683/36)*n^2 + (10223/3)*n + 1024
%F A250853 n=5: a(n) = (763/18)*n^6 + (1949/3)*n^5 + (136493/36)*n^4 + (72691/6)*n^3 + (693923/36)*n^2 + (43319/3)*n + 4096
%F A250853 n=6: a(n) = 198*n^6 + (35807/12)*n^5 + (204911/12)*n^4 + (214827/4)*n^3 + (998209/12)*n^2 + (180451/3)*n + 16384
%F A250853 n=7: a(n) = (15887/18)*n^6 + (39464/3)*n^5 + (2674189/36)*n^4 + (462227/2)*n^3 + (12645859/36)*n^2 + (743119/3)*n + 65536
%e A250853 Some solutions for n=3 k=4
%e A250853 ..2..2..0..0..0....1..2..3..2..2....2..2..1..0..0....3..2..1..1..1
%e A250853 ..0..0..0..0..0....0..1..2..2..3....0..0..0..0..0....0..0..0..0..0
%e A250853 ..1..1..1..1..1....0..1..2..2..3....1..1..1..1..3....0..0..1..1..2
%e A250853 ..0..1..1..1..3....0..1..2..2..3....0..0..0..1..3....0..0..1..1..2
%Y A250853 Row 1 is A001249(n+1)
%K A250853 nonn,tabl
%O A250853 1,1
%A A250853 _R. H. Hardin_, Nov 28 2014