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%I #20 Mar 14 2018 03:54:45
%S 0,0,6,2880,83400,1742400,32810400,600606720,11049696000,207712512000,
%T 4024212192000,80721349632000,1680305519692800,36334168206336000,
%U 816328800967680000,19051455560417280000,461604030434426880000,11603252352344260608000
%N Number of 6-cycles in the n-transposition graph.
%H Robert Israel, <a href="/A300851/b300851.txt">Table of n, a(n) for n = 1..444</a>
%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/TranspositionGraph.html">Transposition Graph</a>
%F a(n) = n!*(binomial(n,3) + 116*binomial(n,4) + 105*binomial(n,5) + 30*binomial(n,6)). - _Andrew Howroyd_, Mar 13 2018
%F a(n) = n!*binomial(n, 3)*(n^3 + 9*n^2 + 16*n - 152)/4.
%F E.g.f.: x^3*(1+113*x-124*x^2+40*x^3)/(1-x)^7. - _Robert Israel_, Mar 14 2018
%p seq(n!*binomial(n, 3)*(n^3 + 9*n^2 + 16*n - 152)/4,n=1..30); # _Robert Israel_, Mar 14 2018
%t Table[n! Binomial[n, 3] (n^3 + 9 n^2 + 16 n - 152)/4, {n, 20}]
%o (PARI) a(n) = n!*(binomial(n,3) + 116*binomial(n,4) + 105*binomial(n,5) + 30*binomial(n,6)); \\ _Andrew Howroyd_, Mar 13 2018
%o (PARI) x='x+O('x^99); concat([0,0], Vec(serlaplace(x^3*(1+113*x-124*x^2+40*x^3)/(1-x)^7))) \\ _Altug Alkan_, Mar 14 2018
%Y Cf. A300843 (4-cycles).
%K nonn
%O 1,3
%A _Eric W. Weisstein_, Mar 13 2018
%E Terms a(9) and beyond from _Andrew Howroyd_, Mar 13 2018