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1/12 of the number of permutations of 3 indistinguishable copies of 1..n with exactly 2 local maxima.
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%I #10 Jul 15 2020 11:40:43

%S 0,1,30,664,13632,274432,5497344,109987840,2199945216,43999756288,

%T 879998926848,17599995314176,351999979683840,7039999912443904,

%U 140799999624609792,2815999998397775872,56319999993188450304,1126399999971143188480,22527999999878130302976

%N 1/12 of the number of permutations of 3 indistinguishable copies of 1..n with exactly 2 local maxima.

%H Andrew Howroyd, <a href="/A152499/b152499.txt">Table of n, a(n) for n = 1..200</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (28,-176,320).

%F a(n) = (11*20^(n-1) - 11*4^(n-1) - 12*(n-1)*4^(n-1))/128. - _Andrew Howroyd_, May 10 2020

%F From _Colin Barker_, Jul 15 2020: (Start)

%F G.f.: x^2*(1 + 2*x) / ((1 - 4*x)^2*(1 - 20*x)).

%F a(n) = 28*a(n-1) - 176*a(n-2) + 320*a(n-3) for n>2.

%F (End)

%o (PARI) a(n) = {(11*20^(n-1) - 11*4^(n-1) - 12*(n-1)*4^(n-1))/128} \\ _Andrew Howroyd_, May 10 2020

%o (PARI) concat(0, Vec(x^2*(1 + 2*x) / ((1 - 4*x)^2*(1 - 20*x)) + O(x^40))) \\ _Colin Barker_, Jul 15 2020

%Y Cf. A152494, A334773.

%K nonn,easy

%O 1,3

%A _R. H. Hardin_, Dec 06 2008

%E Terms a(10) and beyond from _Andrew Howroyd_, May 10 2020