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A180067
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Number of ways to place 9 nonattacking kings on an n X n toroidal board.
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2
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0, 0, 0, 0, 0, 28, 81095, 42752576, 2436444603, 53633024900, 666519047964, 5655962632720, 36502953719310, 191587564345044, 854990702601025, 3346890268570368, 11756179090049177, 37692541754516628, 111774885566128630, 309788198526691600
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OFFSET
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1,6
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LINKS
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FORMULA
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a(n) = 1/362880*n^2 * (n^16 -324*n^14 +46914*n^12 -3975048*n^10 +216203169*n^8 -7756575876*n^6 +179987135516*n^4 -2481599151792*n^2 +15651056776320), n>=10.
G.f.: -x^6*(56520x^22 - 1215064x^21 + 12642984x^20 - 82438064x^19 + 378510176x^18 - 1315100032x^17 + 3593010018x^16 - 7742517098x^15 + 12798616135x^14 - 15614945085x^13 + 14742135008x^12 - 17197088896x^11 + 33440162097x^10 - 55183782403x^9 + 50601858342x^8 - 7249042450x^7 - 32800069391x^6 + 23010354469x^5 + 14572795412x^4 + 1637985772x^3 + 41216559x^2 + 80563x + 28)/(x-1)^19.
General asymptotic formula for number of ways to place k nonattacking kings on an n X n toroidal board: n^2k/k! - 9/2*n^(2k-2)/(k-2)! + (243k+47)*n^(2k-4)/(24*(k-3)!) - (243k^2+141k+80)*n^(2k-6)/(16*(k-4)!) + (98415k^3+114210k^2+140645k+101762)*n^(2k-8)/(5760*(k-5)!)-...
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MATHEMATICA
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CoefficientList[Series[- x^5 (56520 x^22 - 1215064 x^21 + 12642984 x^20 - 82438064 x^19 + 378510176 x^18 - 1315100032 x^17 + 3593010018 x^16 - 7742517098 x^15 + 12798616135 x^14 - 15614945085 x^13 + 14742135008 x^12 - 17197088896 x^11 + 33440162097 x^10 - 55183782403 x^9 + 50601858342 x^8 - 7249042450 x^7 - 32800069391 x^6 + 23010354469 x^5 + 14572795412 x^4 + 1637985772 x^3 + 41216559 x^2 + 80563 x + 28) / (x - 1)^19, {x, 0, 40}], x] (* Vincenzo Librandi, Jun 01 2013 *)
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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