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A179427 Number of ways to place 7 nonattacking kings on an n X n toroidal board.

%I A179427 #17 Aug 05 2024 10:21:51
%S A179427 0,0,0,0,0,3420,576856,19760512,270487188,2209065700,12914201256,
%T A179427 59659859232,231216019632,781647658596,2367858314700,6553746728448,
%U A179427 16815788711212,40446802230372,92003239814224,199311860224800,413589922308360,825997764087012,1594007700404532,2982430581363072,5425904270482500,9622254525739492,16669554533555832,28264133502586912,46982453295836640,76676963241363300
%N A179427 Number of ways to place 7 nonattacking kings on an n X n toroidal board.
%H A179427 Vincenzo Librandi, <a href="/A179427/b179427.txt">Table of n, a(n) for n = 1..1000</a>
%H A179427 V. Kotesovec, <a href="https://oeis.org/wiki/User:Vaclav_Kotesovec">Number of ways of placing non-attacking queens and kings on boards of various sizes</a>
%H A179427 <a href="/index/Rec#order_15">Index entries for linear recurrences with constant coefficients</a>, signature (15, -105, 455, -1365, 3003, -5005, 6435, -6435, 5005, -3003, 1365, -455, 105, -15, 1).
%F A179427 Explicit formula: a(n) = 1/5040*n^2*(n^12 -189*n^10 +15295*n^8 -681135*n^6 +17692024*n^4 -255655596*n^2 +1617230880), n>=8.
%F A179427 G.f.: -4*x^6*(1379*x^16 - 18219*x^15 + 124755*x^14 - 553765*x^13 + 1657983*x^12 - 3369984*x^11 + 4870575*x^10 - 6400905*x^9 + 10992208*x^8 - 19069951*x^7 + 21246441*x^6 - 8631071*x^5 - 7797385*x^4 + 8273322*x^3 + 2866693*x^2 + 131389*x + 855)/(x-1)^15.
%t A179427 CoefficientList[Series[- 4 x^5 (1379 x^16 - 18219 x^15 + 124755 x^14 - 553765 x^13 + 1657983 x^12 - 3369984 x^11 + 4870575 x^10 - 6400905 x^9 + 10992208 x^8 - 19069951 x^7 + 21246441 x^6 - 8631071 x^5 - 7797385 x^4 + 8273322 x^3 + 2866693 x^2 + 131389 x + 855) / (x - 1)^15, {x, 0, 50}], x] (* _Vincenzo Librandi_, Jun 01 2013 *)
%Y A179427 Cf. A179403, A179404, A179424, A179425, A179426.
%K A179427 nonn,easy
%O A179427 1,6
%A A179427 _Vaclav Kotesovec_, Jan 07 2011