A294700 - OEIS

%I #18 Mar 18 2024 12:38:18

%S 1,1,2,6,21,71,218,610,1585,3895,9186,21022,47061,103663,225618,

%T 486626,1042305,2220303,4708714,9949294,20956661,44022231,92253162,

%U 192912386,402622961,838825351,1744789138,3623830750,7516137525,15569193055,32212182306,66571910722

%N Number of permutations of [n] avoiding {1324, 2143, 3421}.

%H Colin Barker, <a href="/A294700/b294700.txt">Table of n, a(n) for n = 0..1000</a>

%H D. Callan, T. Mansour, <a href="http://arxiv.org/abs/1705.00933">Enumeration of small Wilf classes avoiding 1324 and two other 4-letter patterns</a>, arXiv:1705.00933 [math.CO] (2017), Table 1 No 29.

%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (9,-34,70,-85,61,-24,4).

%F G.f.: (1 - 8*x + 27*x^2 - 48*x^3 + 50*x^4 - 30*x^5 + 6*x^6) / ((1 - x)^5*(1 - 2*x)^2).

%F From _Colin Barker_, Nov 10 2017: (Start)

%F a(n) = (1/12)*(12 + 4*(3*2^n-4)*n - 5*n^2 - 2*n^3 - n^4).

%F a(n) = 9*a(n-1) - 34*a(n-2) + 70*a(n-3) - 85*a(n-4) + 61*a(n-5) - 24*a(n-6) + 4*a(n-7) for n>6.

%F (End)

%p p := 1-8*x+27*x^2-48*x^3+50*x^4-30*x^5+6*x^6 ;

%p q := (1-x)^5*(1-2*x)^2 ;

%p taylor(p/q,x=0,40) ;

%p gfun[seriestolist](%) ;

%t LinearRecurrence[{9,-34,70,-85,61,-24,4},{1,1,2,6,21,71,218},40] (* _Harvey P. Dale_, Mar 18 2024 *)

%o (PARI) Vec((1 - 8*x + 27*x^2 - 48*x^3 + 50*x^4 - 30*x^5 + 6*x^6) / ((1 - x)^5*(1 - 2*x)^2) + O(x^40)) \\ _Colin Barker_, Nov 10 2017

%K nonn,easy

%O 0,3

%A _R. J. Mathar_, Nov 07 2017