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%I #9 May 07 2020 11:59:07
%S 0,4,55,396,2114,9528,38637,146080,526240,1831644,6217523,20716164,
%T 68059710,221195824,712856665,2282058360,7266358556,23035517940,
%U 72760054815,229112753980,719545590010,2254604460264,7050252659525,22006821057936,68581455012504,213411502891468
%N Number of permutations of 1..n arranged in a circle with exactly 3 adjacent element pairs in decreasing order.
%C Exactly 2 adjacent element pairs in decreasing order gives A027540(n-1).
%H Andrew Howroyd, <a href="/A151576/b151576.txt">Table of n, a(n) for n = 3..500</a>
%H <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (16,-111,438,-1083,1740,-1817,1190,-444,72).
%F From _Andrew Howroyd_, May 05 2020: (Start)
%F a(n) = n*A000460(n-1).
%F a(n) = n*(3^(n-1) - n*2^(n-1) + n*(n-1)/2).
%F a(n) = 16*a(n-1) - 111*a(n-2) + 438*a(n-3) - 1083*a(n-4) + 1740*a(n-5) - 1817*a(n-6) + 1190*a(n-7) - 444*a(n-8) + 72*a(n-9).
%F G.f.: x^4*(4 - 9*x - 40*x^2 + 131*x^3 - 98*x^4)/((1 - x)^4*(1 - 2*x)^3*(1 - 3*x)^2).
%F (End)
%o (PARI) a(n)={n*(3^(n-1) - n*2^(n-1) + n*(n-1)/2)} \\ _Andrew Howroyd_, May 05 2020
%Y Column k=3 of A334218.
%Y Related sequences: A151577-A151610.
%Y Cf. A000460.
%K nonn,easy
%O 3,2
%A _R. H. Hardin_, May 21 2009
%E Terms a(18) and beyond from _Andrew Howroyd_, May 05 2020
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