%I #11 Aug 14 2017 11:38:04
%S 0,0,24,312,1584,5376,14448,33264,68544,129888,230472,387816,624624,
%T 969696,1458912,2136288,3055104,4279104,5883768,7957656,10603824,
%U 13941312,18106704,23255760,29565120,37234080,46486440,57572424,70770672,86390304,104773056,126295488
%N Number of 5-cycles in the n-triangular 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/JohnsonGraph.html">Johnson Graph</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/TriangularGraph.html">Triangular 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) = 12/5 * binomial(n, 4) * (n^2 + 7*n - 34).
%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.: (24 x^2 (-x^2 - 6 x^3 + 4 x^4))/(-1 + x)^7.
%t Table[12/5 Binomial[n, 4] (n^2 + 7 n - 34), {n, 2, 20}]
%t LinearRecurrence[{7, -21, 35, -35, 21, -7, 1}, {0, 0, 24, 312, 1584, 5376, 14448}, 20]
%t CoefficientList[Series[(24 (-x^2 - 6 x^3 + 4 x^4))/(-1 + x)^7, {x, 0, 20}], x]
%o (PARI) a(n)=12*binomial(n, 4)*(n^2+7*n-34)/5 \\ _Charles R Greathouse IV_, Aug 14 2017
%Y Cf. A002417 (number of 3-cycles in the triangular graph), A151974 (4-cycles), A290940 (6-cycles).
%K nonn,easy
%O 2,3
%A _Eric W. Weisstein_, Aug 14 2017
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