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%I #6 Aug 15 2017 08:41:49
%S 0,0,16,920,7800,36260,122080,334656,794640,1696200,3334320,6137560,
%T 10706696,17859660,28683200,44591680,67393440,99365136,143334480,
%U 202771800,281890840,385759220,520418976,693017600,911950000,1187011800,1529564400,1952712216,2471492520
%N Number of 6-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_08">Index entries for linear recurrences with constant coefficients</a>, signature (8, -28, 56, -70, 56, -28, 8, -1).
%F a(n) = 2*binomial(n, 4) (n^3 + 27*n^2 - 220*n + 392).
%F a(n) = 8*a(n-1) - 28*a(n-2) + 56*a(n-3) - 70*a(n-4) + 56*a(n-5) - 28*a(n-6) + 8*a(n-7) - a(n-8).
%F G.f.: -((4*x^2 (-4*x^2 - 198*x^3 - 222*x^4 + 319*x^5))/(-1 + x)^8).
%t Table[2 Binomial[n, 4] (n^3 + 27 n^2 - 220 n + 392), {n, 2, 20}]
%t LinearRecurrence[{8, -28, 56, -70, 56, -28, 8, -1}, {0, 0, 16, 920, 7800, 36260, 122080, 334656}, 20]
%t CoefficientList[Series[-((4 (-4 x^2 - 198 x^3 - 222 x^4 + 319 x^5))/(-1 + x)^8), {x, 0, 20}], x]
%Y Cf. A002417 (3-cycles), A151974 (4-cycles), A290939 (5-cycles).
%K nonn,easy
%O 2,3
%A _Eric W. Weisstein_, Aug 14 2017
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