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%I #11 Sep 12 2015 11:00:26
%S 0,0,0,976,18510,201528,1232448,5637824,20502396,63720920,174647286,
%T 434439792,997037470,2141831160,4348204020,8412482304,15605151496,
%U 27903377016,48291880442,81188237680,132977239290,212739639640
%N Number of ways to place 5 nonattacking grasshoppers on a toroidal chessboard of size n x n.
%C The Grasshopper moves on the same lines as a queen, but must jump over a hurdle to land on the square immediately beyond.
%H Vincenzo Librandi, <a href="/A190400/b190400.txt">Table of n, a(n) for n = 1..1000</a>
%H V. Kotesovec, <a href="https://oeis.org/wiki/User:Vaclav_Kotesovec">Number of ways of placing non-attacking queens, kings, bishops and knights</a> (in English and Czech)
%F Explicit formula: a(n) = 1/120*n^2*(n^8 -10*n^6 -240*n^5 +995*n^4 +640*n^3 +1870*n^2 -41680*n +69624) + 2*n^2*(n-3)*(2n+1)*(-1)^n, n>6.
%F G.f.: 2*x^4*(144*x^18 -874*x^17 +1356*x^16 +2195*x^15 -8778*x^14 +4282*x^13 +16170*x^12 -23696*x^11 -5686*x^10 +36079*x^9 -33008*x^8 -33909*x^7 -13310*x^6 -61448*x^5 -197358*x^4 -109070*x^3 -50114*x^2 -6327*x -488)/((x-1)^11*(x+1)^5).
%t CoefficientList[Series[2 x^3 (144 x^18 - 874 x^17 + 1356 x^16 + 2195 x^15 - 8778 x^14 + 4282 x^13 + 16170 x^12 - 23696 x^11 - 5686 x^10 + 36079 x^9 - 33008 x^8 - 33909 x^7 - 13310 x^6 - 61448 x^5 - 197358 x^4 - 109070 x^3 - 50114 x^2 - 6327 x- 488) / ((x - 1)^11 (x + 1)^5), {x, 0, 50}], x] (* _Vincenzo Librandi_, Jun 03 2013 *)
%Y Cf. A190397, A190398, A190399, A173775.
%K nonn,easy
%O 1,4
%A _Vaclav Kotesovec_, May 10 2011
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