A294797 - OEIS

%I #12 Mar 21 2021 21:22:05

%S 1,1,2,6,21,73,237,711,1988,5253,13301,32673,78669,187230,443398,

%T 1050209,2497187,5976456,14419577,35101838,86229874,213707024,

%U 534022585,1344450284,3407215499,8684458421,22243751970,57208235104,147636483121,382077767531,991081141309,2575597009441

%N Number of permutations of [n] avoiding {1243, 1324, 3412}.

%H Colin Barker, <a href="/A294797/b294797.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 72.

%H <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (13,-74,243,-510,715,-678,429,-173,40,-4).

%F From _Colin Barker_, Nov 23 2017: (Start)

%F G.f.: (1 - 12*x + 63*x^2 - 189*x^3 + 358*x^4 - 447*x^5 + 367*x^6 - 192*x^7 + 64*x^8 - 10*x^9) / ((1 - x)^6*(1 - 2*x)^2*(1 - 3*x + x^2)).

%F a(n) = (1/120)*(3*2^(2-n)*(5*2^(2+n)*(-1+2^n) - (-5+sqrt(5))*(3+sqrt(5))^n + (3-sqrt(5))^n*(5+sqrt(5))) + 4*(-73+15*2^(1+n))*n - 120*n^2 - 65*n^3 - 3*n^5).

%F a(n) = 13*a(n-1) - 74*a(n-2) + 243*a(n-3) - 510*a(n-4) + 715*a(n-5) - 678*a(n-6) + 429*a(n-7) - 173*a(n-8) + 40*a(n-9) - 4*a(n-10) for n>9.

%F (End)

%p 1+x*(1 -11*x +54*x^2 -152*x^3 +268*x^4 -311*x^5 +237*x^6 -109*x^7 +30*x^8 -4*x^9) /((1-x)^6 *(1-2*x)^2 *(1-3*x+x^2)) ;

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

%p gfun[seriestolist](%) ;

%o (PARI) Vec((1 - 12*x + 63*x^2 - 189*x^3 + 358*x^4 - 447*x^5 + 367*x^6 - 192*x^7 + 64*x^8 - 10*x^9) / ((1 - x)^6*(1 - 2*x)^2*(1 - 3*x + x^2)) + O(x^40)) \\ _Colin Barker_, Nov 23 2017

%K nonn,easy

%O 0,3

%A _R. J. Mathar_, Nov 09 2017