Column 1 and Row 1
Number of (n+2)X3 0..2 arrays with each 3X3 subblock having rows and columns in lexicographically nondecreasing order

Empirical: a(n) = (1/362880)*n^9
                + (1/1008)*n^8
                + (431/12096)*n^7
                + (373/720)*n^6
                + (82453/17280)*n^5
                + (637/18)*n^4
                + (2819461/18144)*n^3
                + (119729/315)*n^2
                + (605807/1260)*n
                + 112

Column 2 and Row 2
Number of (n+2)X4 0..2 arrays with each 3X3 subblock having rows and columns in lexicographically nondecreasing order

Empirical: a(n) = (1/9979200)*n^11
                + (29/3628800)*n^10
                + (257/241920)*n^9
                + (4621/120960)*n^8
                + (400927/604800)*n^7
                + (1134257/172800)*n^6
                + (31386443/725760)*n^5
                + (20355493/90720)*n^4
                + (80111509/100800)*n^3
                + (40459121/25200)*n^2
                + (7603747/4620)*n
                + 273

Column 3 and Row 3
Number of (n+2)X5 0..2 arrays with each 3X3 subblock having rows and columns in lexicographically nondecreasing order

Empirical: a(n) = (1/444787200)*n^13
                + (107/479001600)*n^12
                + (11/1088640)*n^11
                + (3623/6220800)*n^10
                + (78227/3628800)*n^9
                + (6420917/14515200)*n^8
                + (148481/27216)*n^7
                + (1861160843/43545600)*n^6
                + (4966108109/21772800)*n^5
                + (1448539279/1555200)*n^4
                + (3663351401/1330560)*n^3
                + (2013261953/415800)*n^2
                + (15650889/3640)*n
                + 556

Column 4 and Row 4
Number of (n+2)X6 0..2 arrays with each 3X3 subblock having rows and columns in lexicographically nondecreasing order

Empirical: a(n) = (1/27243216000)*n^15
                + (97/21794572800)*n^14
                + (1549/6227020800)*n^13
                + (4127/479001600)*n^12
                + (676031/2395008000)*n^11
                + (54763/6220800)*n^10
                + (60678553/304819200)*n^9
                + (904333037/304819200)*n^8
                + (910283537/31104000)*n^7
                + (8446896821/43545600)*n^6
                + (213392977159/239500800)*n^5
                + (52396736149/17107200)*n^4
                + (70560211894163/9081072000)*n^3
                + (1840763445463/151351200)*n^2
                + (1738288907/180180)*n
                + 1019

Column 5 and Row 5
Number of (n+2)X7 0..2 arrays with each 3X3 subblock having rows and columns in lexicographically nondecreasing order

Empirical: a(n) = (1/2155681382400)*n^17
                + (47/697426329600)*n^16
                + (611/134120448000)*n^15
                + (33559/174356582400)*n^14
                + (189341/33210777600)*n^13
                + (178309/1277337600)*n^12
                + (147904279/44706816000)*n^11
                + (87015563/1219276800)*n^10
                + (11642696389/9754214400)*n^9
                + (69578815777/4877107200)*n^8
                + (3848584871/32256000)*n^7
                + (44384094767/63866880)*n^6
                + (247986427893683/87178291200)*n^5
                + (124742663704661/14529715200)*n^4
                + (1115145293971/58212000)*n^3
                + (2731472131849/100900800)*n^2
                + (9965552297/510510)*n
                + 1735

Column 6 and Row 6
Number of (n+2)X8 0..2 arrays with each 3X3 subblock having rows and columns in lexicographically nondecreasing order

Empirical: a(n) = (1/212666259456000)*n^19
                + (67/83147710464000)*n^18
                + (12559/194011324416000)*n^17
                + (12833/3923023104000)*n^16
                + (333169/2853107712000)*n^15
                + (196437919/62768369664000)*n^14
                + (1153058911/17118646272000)*n^13
                + (1339332949/1034643456000)*n^12
                + (28852162981/1207084032000)*n^11
                + (350922659539/877879296000)*n^10
                + (4735468557467/877879296000)*n^9
                + (11975048873639/219469824000)*n^8
                + (131882024236729/329204736000)*n^7
                + (7064899018536199/3362591232000)*n^6
                + (1862906706349061/237758976000)*n^5
                + (13955757767803297/653837184000)*n^4
                + (26671169520943/623750400)*n^3
                + (851721864110749/15437822400)*n^2
                + (4258571767339/116396280)*n
                + 2793

Column 7 and Row 7
Number of (n+2)X9 0..2 arrays with each 3X3 subblock having rows and columns in lexicographically nondecreasing order

Empirical: a(n) = (1/25519951134720000)*n^21
                + (1/127919554560000)*n^20
                + (1877/2551995113472000)*n^19
                + (101861/2328135892992000)*n^18
                + (10763959/5820339732480000)*n^17
                + (24725641/418455797760000)*n^16
                + (836589317/564915326976000)*n^15
                + (22655691343/753220435968000)*n^14
                + (5887296211331/11298306539520000)*n^13
                + (436620904451/52672757760000)*n^12
                + (3650470717199/28970016768000)*n^11
                + (2036750095277/1170505728000)*n^10
                + (1572848084986607/79009136640000)*n^9
                + (1160407108626427/6584094720000)*n^8
                + (2296534278781759/1975228416000)*n^7
                + (525799308273622787/94152554496000)*n^6
                + (25696337451699745273/1333827855360000)*n^5
                + (2682952836795541619/55576160640000)*n^4
                + (124299364108443341/1407929402880)*n^3
                + (73695052155737/701719200)*n^2
                + (1503827501467/23279256)*n
                + 4299

Column 8 and Row 8
Number of (n+2)X10 0..2 arrays with each 3X3 subblock having rows and columns in lexicographically nondecreasing order

Empirical: a(n) = (1/3655545353349120000)*n^23
                + (97/1543957043650560000)*n^22
                + (43/6288115959595008)*n^21
                + (37903/80205560709120000)*n^20
                + (4363319/187146308321280000)*n^19
                + (111930887/128047474114560000)*n^18
                + (173896301/6722492391014400)*n^17
                + (999595589/1614043791360000)*n^16
                + (966949557157/79088145776640000)*n^15
                + (2303051708719/11298306539520000)*n^14
                + (762601942063/254936147558400)*n^13
                + (260118396674119/6373403688960000)*n^12
                + (6455418497147533/12167407042560000)*n^11
                + (10938624786824941/1738201006080000)*n^10
                + (15334472267219503/243348140851200)*n^9
                + (54278261336966987/108637562880000)*n^8
                + (9086167557375919/3017710080000)*n^7
                + (11925742307499655463/889218570240000)*n^6
                + (6823870134571712569/157688093122560)*n^5
                + (7908447122576853679/78218300160000)*n^4
                + (17170767258622048961/100380151872000)*n^3
                + (406964817911133233/2151003254400)*n^2
                + (2131730921383/19612560)*n
                + 6377