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A172135 Number of ways to place 4 nonattacking knights on an n X n board.

%I A172135 #20 Apr 23 2022 18:41:19
%S A172135 0,1,18,412,4436,26133,111066,376560,1080942,2732909,6253408,13204356,
%T A172135 26100160,48819677,87137934,149398608,247349946,397168485,620696612,
%U A172135 946921684,1413726108,2069939461,2977725410,4215337872
%N A172135 Number of ways to place 4 nonattacking knights on an n X n board.
%D A172135 E. Bonsdorff, K. Fabel, O. Riihimaa, Schach und Zahl, 1966, p. 51-63
%H A172135 Vincenzo Librandi, <a href="/A172135/b172135.txt">Table of n, a(n) for n = 1..1000</a>
%H A172135 Vaclav 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 A172135 <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (9,-36,84,-126,126,-84,36,-9,1).
%F A172135 a(n) = (n^8 - 54*n^6 + 144*n^5 + 1019*n^4 - 5232*n^3 - 2022*n^2 + 51120*n - 77184)/24, n >= 6. (Karl Fabel, 1966)
%F A172135 G.f.: x^2 * ( 1 + 9*x + 286*x^2 + 1292*x^3 - 345*x^4 +3099*x^5 - 5142*x^6 + 3606*x^7 - 1162*x^8 - 390*x^9 + 690*x^10 - 312*x^11 + 48*x^12) / (1-x)^9. - _Vaclav Kotesovec_, Mar 25 2010
%F A172135 E.g.f.: x^2/2! + 18*x^3/3! + 412*x^4/4! + 4436*x^5/5! + (1/120)*(385920 + 161040*x + 17940*x^2 - 1200*x^3 - 2660*x^4 - 4484*x^5 + (-385920 + 224880*x - 49860*x^2 + 2940*x^3 + 3250*x^4 + 1920*x^5 + 1060*x^6 + 140*x^7 + 5*x^8)*exp(x)). - _G. C. Greubel_, Apr 19 2022
%t A172135 CoefficientList[Series[x*(1 +9*x +286*x^2 +1292*x^3 -345*x^4 +3099*x^5 -5142*x^6 +3606*x^7 -1162*x^8 -390*x^9 +690*x^10 -312*x^11 +48*x^12)/(1-x)^9, {x, 0, 40}], x] (* _Vincenzo Librandi_, May 26 2013 *)
%o A172135 (Magma) [0,1,18,412,4436] cat [(n^8 -54*n^6 +144*n^5 +1019*n^4 -5232*n^3 -2022*n^2 +51120*n -77184)/24: n in [6..50]]; // _G. C. Greubel_, Apr 19 2022
%o A172135 (SageMath) [0,1,18,412,4436] + [(n^8 -54*n^6 +144*n^5 +1019*n^4 -5232*n^3 -2022*n^2 +51120*n -77184)/24 for n in (6..50)] # _G. C. Greubel_, Apr 19 2022
%Y A172135 Cf. A061994, A172127, A172132, A172134.
%Y A172135 Column k=4 of A244081.
%K A172135 nonn,easy
%O A172135 1,3
%A A172135 _Vaclav Kotesovec_, Jan 26 2010