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A288962 Number of 4-cycles in the n X n rook graph.

%I A288962 #12 Feb 16 2025 08:33:48
%S A288962 0,1,9,60,250,765,1911,4144,8100,14625,24805,39996,61854,92365,133875,
%T A288962 189120,261256,353889,471105,617500,798210,1018941,1285999,1606320,
%U A288962 1987500,2437825,2966301,3582684,4297510,5122125,6068715,7150336,8380944,9775425,11349625,13120380,15105546,17324029,19795815,22542000
%N A288962 Number of 4-cycles in the n X n rook graph.
%H A288962 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GraphCycle.html">Graph Cycle</a>
%H A288962 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RookGraph.html">Rook Graph</a>
%H A288962 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (6, -15, 20, -15, 6, -1).
%F A288962 a(n) = n*binomial(n,2)*(n^2-4*n+5)/2.
%F A288962 a(n) = 6*a(n-1)-15*a(n-2)+20*a(n-3)-15*a(n-4)+6*a(n-5)-a(n-6).
%F A288962 G.f.: (x^2*(1+3*x+21*x^2+5*x^3))/(-1+x)^6.
%t A288962 Table[n^2 (n - 1) (n^2 - 4 n + 5)/4, {n, 20}]
%t A288962 Table[n Binomial[n, 2] (n^2 - 4 n + 5)/2, {n, 20}]
%t A288962 LinearRecurrence[{6, -15, 20, -15, 6, -1}, {0, 1, 9, 60, 250, 765}, 20]
%t A288962 CoefficientList[Series[(x (1 + 3 x + 21 x^2 + 5 x^3))/(-1 + x)^6, {x, 0, 20}], x]
%o A288962 (Magma) [n^2*(n-1)*(n^2-4*n+5)/4 : n in [1..50]]; // _Wesley Ivan Hurt_, Apr 23 2021
%Y A288962 Cf. A288961 (3-cycles), A288963 (5-cycles), A288960 (6-cycles).
%K A288962 nonn,easy
%O A288962 1,3
%A A288962 _Eric W. Weisstein_, Jun 20 2017