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%I #8 Dec 24 2016 09:39:53
%S 0,0,0,0,22650,4987800,240023070,5219088000,68483325960,630486309600,
%T 4456523194200,25647802519680,125166919041450,533442526857240,
%U 2029603476250350,7011735609715200,22291042191643680,65914292362262400,182880685655641440,479548000781222400
%N Number of ways to place 9 points on an n X n board so that no more than 2 points are on a vertical or horizontal straight line.
%C Column 9 of triangle A279445.
%C Rotations and reflections of placements are counted.
%H Heinrich Ludwig, <a href="/A279443/b279443.txt">Table of n, a(n) for n = 1..1000</a>
%H <a href="/index/Rec#order_19">Index entries for linear recurrences with constant coefficients</a>, signature (19,-171,969,-3876,11628, -27132,50388,-75582,92378,-92378,75582,-50388,27132,-11628,3876, -969,171,-19,1).
%F a(n) = (n^18 -204*n^16 +1260*n^15 +6846*n^14 -104076*n^13 +394504*n^12 +128520*n^11 -6237075*n^10 +24018372*n^9 -43820196*n^8 +30400020*n^7 +34251148*n^6 -99199296*n^5 +98504496*n^4 -47779200*n^3 +9434880*n^2)/362880; factorized: a(n) = n^2*(n-1)^2*(n-2)^2*(n-3)^2*(n-4)^2*(n^8 +20*n^7 +26*n^6 -820*n^5 -247*n^4 +9704*n^3 -9104*n^2 -14700*n +16380)/9!.
%F a(n) = SUM(1<=j<=19, C(19,j)*(-1)^(j-1)*a(n-j)).
%F G.f.: 30*x^5*(755 +151915*x +4970934*x^2 +49653854*x^3 +187307071*x^4 +275138271*x^5 +119386656*x^6 -31251744*x^7 -19595619*x^8 +1706821*x^9 +667466*x^10 -26334*x^11 -2543*x^12 +17*x^13) / (1 -x)^19. - _Colin Barker_, Dec 24 2016
%o (PARI) concat(vector(4), Vec(30*x^5*(755 +151915*x +4970934*x^2 +49653854*x^3 +187307071*x^4 +275138271*x^5 +119386656*x^6 -31251744*x^7 -19595619*x^8 +1706821*x^9 +667466*x^10 -26334*x^11 -2543*x^12 +17*x^13) / (1 -x)^19 + O(x^30))) \\ _Colin Barker_, Dec 24 2016
%Y Cf. A279444, A279445, A197458.
%Y Same problem but 2..8 points: A083374, A279437, A279438, A279439, A279440, A279441, A279442.
%K nonn,easy
%O 1,5
%A _Heinrich Ludwig_, Dec 24 2016
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