%I #10 Dec 23 2014 16:02:52

%S 0,0,0,28565107560,272454472598400,142575583958069520,

%T 18434332194257819520,1017379423994278214520,31340171098354199988480,

%U 628988614673638882511520,9076042310340825810167040,100702084869046006909196040,901495065442530655732775040

%N Number of ways of n-coloring the square grid graph G_(5,5) such that no rectangle exists with sides parallel to the axes having all 4 corners of the same color.

%C The square grid graph G_(5,5) has 25 vertices, 40 edges and 100 rectangles with sides parallel to the axes.

%C a(4) = A200045(5,5) = 272454472598400.

%H Alois P. Heinz, <a href="/A252839/b252839.txt">Table of n, a(n) for n = 0..1000</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GridGraph.html">Grid Graph</a>

%F a(n) = n *(n-1) *(n-2) *(n^22 +3*n^21 +7*n^20 -85*n^19 -269*n^18 -237*n^17 +4177*n^16 +8805*n^15 -12139*n^14 -109047*n^13 -9663*n^12 +503305*n^11 +477366*n^10 -1657912*n^9 -986868*n^8 -206320*n^7 +7988176*n^6 -5442232*n^5 -2404368*n^4 -428800*n^3 +3512736*n^2 -1855392*n +585552).

%F G.f.: 120 *x^3 *(917131*x^22 +256115122*x^21 +86195456293*x^20 +29895128202300*x^19 +4101355164443025*x^18 +240560314496910906*x^17 +6809815211785690527*x^16 +102308317235278417008*x^15 +873828591412946507550*x^14 +4453329070294962830500*x^13 +13994509523514621306946*x^12 +27682853019612987461512*x^11 +34828722027105722198418*x^10 +27894388401732252010500*x^9 +14097481092846476345550*x^8 +4408206551672480573616*x^7 +824806705925084182407*x^6 +87606990289152898218*x^5 +4864299142912914025*x^4 +123465337047190300*x^3 +1129175427753901*x^2 +2264264831682*x +238042563) / (x-1)^26.

%p a:= n-> (((((((((((((((((((((n^3-100)*n^2+400)*n+4350)*n-4200)*n

%p -30200)*n-55020)*n+293200)*n+314200)*n-1051875)*n-2083400)*n

%p +4941600)*n-561540)*n+6633400)*n-29819400)*n+29898680)*n

%p -4100160)*n-9600)*n-13251200)*n+13177200)*n-5467440)*n+1171104)*n:

%p seq(a(n), n=0..30);

%Y Cf. A200045, A252778, A252779, A252780.

%K nonn,easy

%O 0,4

%A _Alois P. Heinz_, Dec 22 2014