Column 1 and Row 1
Number of (n+2)X3 0..7 arrays with each 3X3 subblock having rows and columns in lexicographically nondecreasing order

Empirical: a(n) = (1/55745857612076058813234317117419771556272886109483581752463927935846946310374691578057284710599874844234646982443450754604453404911734348832487342619913750049708004343808000000000000000000000000000)*n^119
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                + (307196017630686704941639867635061754365611715610778071/703894000240227142498903562780766092633745607200)*n
                + 32782

Column 2 and Row 2
Number of (n+2)X4 0..7 arrays with each 3X3 subblock having rows and columns in lexicographically nondecreasing order

Empirical: a(n) = (9009697/11860866214400234428385736525697085402851042986904022830918688585026248848891656728613624772038243157419723543093593637659700200918506977662589657826188464498032561660595449301565440000000000000000000000000000000)*n^126
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                + (6556496811679/1902756144295478060109060287690699453258050869016869960949269400633040636706071897986203902175383657306940134881796877173938084793172160006803682177869839167567739171635200000000000000000000000000)*n^120
                + (48366414161498809/10468705654849964096382031383553008742962044339809123333796042804853886631056280108555358631098568045865661405153699672226188432847274056118776965750218544610273803632640000000000000000000000000000)*n^119
                + (1796612810401331903/350462413495418829008773198645565875898939974694176226495879706789511494797200381947439639845303469720882128049009746792160722196596158665961457795811263334816363315200000000000000000000000000000)*n^118
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                + (24663619628290233610704211/6251171110870593697734948407841062715649707794775814107681891262334763493351909581997869514409920658375242142461875391243556020226670559343318679514824287852093717348352000000000000000000000000000)*n^116
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                + (6367773441105597370960253587/11761893403796104790647601307734065772658868157396532797118715454220099266123648931899615493936237305147714216125888457960276516089859403490154766312213967798272000000000000000000000000000)*n^112
                + (81798815034483280739934349659373/318201223334841049246984213950305529385324736758138342636336677020418756931738002354069954880596419951764054239834303818032480747788160647992579838625074307399680000000000000000000000000000)*n^111
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                + (643132409874480909006280416717819027206992765417796500921871568905514326835133453584990331023087999404422426122485699854975971866149685536260501253516599743/368601365620015606510506987198291440420308182836487622246833442713532267285324135345364881646938937337151746175622624521225733605144267849728000000000000000000000000000)*n^32
                + (9944213531972705885486172075051503152714563135982468169917350312888086223216988059093305796603712079696353218898252135234156823235666504546062245650120679/441333052705957383274074457852360441116269376001541693303200961103367178263079663967151438753518842597164447049356590662387133147921776640000000000000000000000000000)*n^31
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                + (3301907599379186707774287493269516424554903875190608104951/365096131096823369953077132667329301797322634734500)*n
                + 385162

Column 3 and Row 3
Number of (n+2)X5 0..7 arrays with each 3X3 subblock having rows and columns in lexicographically nondecreasing order

Empirical: a(n) = (394465919299/90137618550950650236875811586745746052580662127606486669750413858133548887788337905382315907100435404259353436860363861615304523448028488106292623927983646917075369775751265613441839412585431040000000000000000000000000000000)*n^133
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                + (2919808862266657/21862143718396956157379532279103988855828440972012245129699348498213327404265907811152635437084752705374570321819152040168640437411600409436403740947849538422768704772192885183953878101524480000000000000000000000000000000)*n^131
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                + (7872914165442542701/9475246138051976076933779334119392312315544978502636789708355051933746463275496213205431571031895169121881273612304801683693392049878543362253110328878109933532279841428004371872748666880000000000000000000000000000000)*n^129
                + (591629717812665522079/361519202214919145377197249303247163078899790240834615886401628071600064914217697088143283051725651438153173593492734075867662123996092679155732770542224397900032479414949294711714611200000000000000000000000000000000)*n^128
                + (2264508672254736461594093/809803012961418885644921838439273645296735530139469539585539646880384145407847641477440954035865459221463108849423724329943563157751247601308841406014582651296072753889486420154240729088000000000000000000000000000000)*n^127
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                + (4264237465883133210343407593/828104113878125458272749604703214689944509183085662684922322984845469010540799306143205802265942794990758880099625446702059068573219396258624441564592067339499000668667027733054750720000000000000000000000000000000)*n^125
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                + (31841940844178295845341515553477010620641683483991989842305677/29196957689027351244523042308720754153847662312054281683719734080111155062735348894059258276953873450035714211619467453891192854747189497391062444447425992264001781760000000000000000000000000000000)*n^106
                + (32055522921385666204592763273994138571713302943828982436367/132997784131245042079173692180916861512949661509806886325604001081186658753090481519693905156425701664850198879259821336343623795649082902057189805316594376638464000000000000000000000000000000)*n^105
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