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%I #7 Aug 02 2023 17:17:14
%S 0,0,0,6,696,6720,39840,184800,736512,2644992,8801280,27624960,
%T 82790400,238977024,668688384,1822679040,4858183680,12700876800,
%U 32647938048,82682707968,206650736640,510425825280,1247438438400,3019527684096,7245593051136,17248655769600
%N Number of 8-cycles in the hypercube graph Q_n.
%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/HypercubeGraph.html">Hypercube Graph</a>
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (10,-40,80,-80,32).
%F a(n) = 2^(n - 4)*n*(n - 1)*(n - 2)*(27*n - 79).
%F a(n) = 10*a(n-1) - 40*a(n-2) + 80*a(n-3) - 80*a(n-4) + 32*a(n-5).
%F G.f.: -6*x^3*(1 + 106*x)/(-1 + 2*x)^5.
%t Table[Length[FindCycle[HypercubeGraph[n], {8}, All]], {n, 0, 9}]
%t Table[2^(n - 4) n (n - 1) (n - 2) (27 n - 79), {n, 0, 20}]
%t Table[3 2^(n - 3) Binomial[n, 3] (27 n - 79), {n, 0, 20}]
%t LinearRecurrence[{10, -40, 80, -80, 32}, {0, 0, 0, 6, 696}, 20]
%t CoefficientList[Series[6 x^3 (1 + 106 x)/(1 - 2 x)^5, {x, 0, 20}], x]
%Y Cf. A001788 (4-cycles).
%Y Cf. A290031 (6-cycles).
%K nonn
%O 0,4
%A _Eric W. Weisstein_, Aug 02 2023
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