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%I #7 Jun 22 2017 11:38:00
%S 0,0,36,288,1320,4464,12348,29568,63504,125280,230868,402336,669240,
%T 1070160,1654380,2483712,3634464,5199552,7290756,10041120,13607496,
%U 18173232,23951004,31185792,40158000,51186720,64633140,80904096,100455768,123797520
%N Number of 5-cycles in the n X n rook 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/RookGraph.html">Rook Graph</a>
%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (7, -21, 35, -35, 21, -7, 1).
%F a(n) = 6*n*binomial(n,3)*(n^2-2*n+7)/5.
%F a(n) = 7*a(n-1)-21*a(n-2)+35*a(n-3)-35*a(n-4)+21*a(n-5)-7*a(n-6)+a(n-7).
%F G.f.: (-12*x^3*(3+3*x+5*x^2+x^3))/(-1+x)^7.
%t Table[(n - 2) (n - 1) n^2 (n^2 - 2 n + 7)/5, {n, 20}]
%t Table[6 n Binomial[n, 3] (n^2 - 2 n + 7)/5, {n, 20}]
%t LinearRecurrence[{7, -21, 35, -35, 21, -7, 1}, {0, 0, 36, 288, 1320, 4464, 12348}, 20]
%t CoefficientList[Series[-((12 x^2 (3 + 3 x + 5 x^2 + x^3))/(-1 + x)^7), {x, 0, 20}], x]
%Y Cf. A288961 (3-cycles), A288962 (4-cycles), A288960 (6-cycles).
%K nonn,easy
%O 1,3
%A _Eric W. Weisstein_, Jun 20 2017
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