%I #11 Sep 13 2017 11:12:14

%S 0,0,9,576,6900,44100,196245,686784,2023056,5232600,12224025,26310240,

%T 52936884,100663836,182452725,317318400,532407360,865571184,

%U 1368508041,2110550400,3183182100,4705372980,6829824309,9750223296,13709610000,19009965000,26023131225

%N Number of 4-cycles in the n X n rook complement graph.

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

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

%H <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 a(n) = (n-2)*(n-1)^2*n^2*(-4 + 5*n - 4*n^2 + n^3)/8.

%F a(n) = 9*a(n-1) - 36*a(n-2) + 84*a(n-3) - 126*a(n-4) + 126*a(n-5) - 84*a(n-6) + 36*a(n-7) - 9*a(n-8) + a(n-9).

%F G.f.: -((3 x^3 (3 + 165 x + 680 x^2 + 660 x^3 + 165 x^4 + 7 x^5))/(-1 + x)^9).

%t Table[(-2 + n) (-1 + n)^2 n^2 (-4 + 5 n - 4 n^2 + n^3)/8, {n, 20}]

%t LinearRecurrence[{9, -36, 84, -126, 126, -84, 36, -9, 1}, {0, 0, 9, 576, 6900, 44100, 196245, 686784, 2023056}, 30]

%t CoefficientList[Series[-((3 x^2 (3 + 165 x + 680 x^2 + 660 x^3 + 165 x^4 + 7 x^5))/(-1 + x)^9), {x, 0, 20}], x]

%Y Cf. A179058 (3-cycles), A291911 (5-cycles), A291912 (6-cycles).

%K nonn

%O 1,3

%A _Eric W. Weisstein_, Sep 05 2017