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a(n) = 3*(n - 1)^2*n^3.
0

%I #10 Feb 16 2025 08:33:53

%S 0,0,24,324,1728,6000,16200,37044,75264,139968,243000,399300,627264,

%T 949104,1391208,1984500,2764800,3773184,5056344,6666948,8664000,

%U 11113200,14087304,17666484,21938688,27000000,32955000,39917124,48009024,57362928,68121000

%N a(n) = 3*(n - 1)^2*n^3.

%C Also the number of 5-cycles in the complete tripartite graph K_{n,n,n} for n >= 1.

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GraphCycle.html">Graph Cycle</a>

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CompleteTripartiteGraph.html">Complete Tripartite Graph</a>

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (6,-15,20,-15,6,-1).

%F G.f.: 12*x^2*(2 + 15*x + 12*x^2 + x^3)/(x - 1)^6.

%F 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 a(n) = 3*A099762(n-1).

%F a(n) = 3*(n - 1)^5 + 9*(n - 1)^4 + 9*(n - 1)^3 + 3*(n - 1)^2.

%t Table[3 (n - 1)^2 n^3, {n, 0, 20}]

%t LinearRecurrence[{6, -15, 20, -15, 6, -1}, {0, 0, 24, 324, 1728, 6000}, 20]

%t CoefficientList[Series[12 x^2 (2 + 15 x + 12 x^2 + x^3)/(x - 1)^6, {x, 0, 20}], x]

%o (PARI) a(n) = 3*(n-1)^2*n^3; \\ _Altug Alkan_, Mar 13 2018

%K nonn,easy,changed

%O 0,3

%A _Eric W. Weisstein_, Mar 13 2018