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A116725 Number of permutations of length n which avoid the patterns 132, 3421, 4231.

%I A116725 #16 Feb 07 2022 00:27:53
%S A116725 1,2,5,12,26,52,99,184,340,632,1189,2268,4382,8556,16839,33328,66216,
%T A116725 131888,263113,525428,1049906,2098692,4196075,8390632,16779516,
%U A116725 33557032,67111789,134221004,268439110,536874972,1073746319,2147488608,4294972752,8589940576
%N A116725 Number of permutations of length n which avoid the patterns 132, 3421, 4231.
%H A116725 Colin Barker, <a href="/A116725/b116725.txt">Table of n, a(n) for n = 1..1000</a>
%H A116725 Lara Pudwell, <a href="http://faculty.valpo.edu/lpudwell/maple/webbook/bookmain.html">Systematic Studies in Pattern Avoidance</a>, 2005.
%H A116725 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (6,-14,16,-9,2).
%F A116725 G.f.: x*(1 - 4*x + 7*x^2 - 6*x^3 + x^4) / ((1 - x)^4*(1 - 2*x)).
%F A116725 Row sums of triangle A134398. Also, binomial transform of [1, 1, 2, 2, 1, 1, 1, ...]. - _Gary W. Adamson_, Oct 23 2007
%F A116725 From _Colin Barker_, Oct 20 2017: (Start)
%F A116725 a(n) = (3*2^n + 2*n - 3*n^2 + n^3)/6.
%F A116725 a(n) = 6*a(n-1) - 14*a(n-2) + 16*a(n-3) - 9*a(n-4) + 2*a(n-5) for n>5.
%F A116725 (End)
%p A116725 seq((3*2^n +2*n -3*n^2 +n^3)/6, n=1..40); # _G. C. Greubel_, Nov 29 2019
%t A116725 Table[(3*2^n +2*n -3*n^2 +n^3)/6, {n, 40}] (* _G. C. Greubel_, Nov 29 2019 *)
%o A116725 (PARI) Vec(x*(1 -4*x +7*x^2 -6*x^3 +x^4)/((1-x)^4*(1-2*x)) + O(x^40)) \\ _Colin Barker_, Oct 20 2017
%o A116725 (PARI) vector(40, n, (3*2^n +2*n -3*n^2 +n^3)/6) \\ _G. C. Greubel_, Nov 29 2019
%o A116725 (Magma) [(3*2^n +2*n -3*n^2 +n^3)/6: n in [1..40]]; // _G. C. Greubel_, Nov 29 2019
%o A116725 (Sage) [(3*2^n +2*n -3*n^2 +n^3)/6 for n in (1..40)] # _G. C. Greubel_, Nov 29 2019
%o A116725 (GAP) List([1..40], n-> (3*2^n +2*n -3*n^2 +n^3)/6); # _G. C. Greubel_, Nov 29 2019
%Y A116725 Cf. A134398.
%K A116725 nonn,easy
%O A116725 1,2
%A A116725 _Lara Pudwell_, Feb 26 2006