A294770 - OEIS

%I #9 Dec 24 2023 12:13:54

%S 1,1,2,6,21,75,253,774,2130,5314,12169,25895,51756,98034,177282,

%T 307933,516327,839223,1326868,2046700,3087767,4565949,6630075,9469032,

%U 13319968,18477696,25305411,34246837,45839926,60732236,79698120,103657863,133698909,171099325,217353654,274201314,343657705

%N Number of permutations of [n] avoiding {4231, 4123, 1234}.

%H Colin Barker, <a href="/A294770/b294770.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 2 No 144.

%H <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (9,-36,84,-126,126,-84,36,-9,1).

%F G.f.: (1 - 8*x + 29*x^2 - 60*x^3 + 81*x^4 - 66*x^5 + 40*x^6 - 15*x^7 + 3*x^8) / (1 - x)^9.

%F From _Colin Barker_, Nov 11 2017: (Start)

%F a(n) = (40320 - 22704*n + 33868*n^2 - 16996*n^3 + 6405*n^4 - 616*n^5 + 42*n^6 - 4*n^7 + 5*n^8) / 40320.

%F a(n) = 9*a(n-1) - 36*a(n-2) + 84*a(n-3) - 126*a(n-4) + 126*a(n-5) - 84*a(n-6) + 36*a(n-7) - 9*a(n-8) + a(n-9) for n>8.

%F (End)

%p -(3*x^8-15*x^7+40*x^6-66*x^5+81*x^4-60*x^3+29*x^2-8*x+1)/(x-1)^9 ;

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

%p gfun[seriestolist](%) ;

%t LinearRecurrence[{9,-36,84,-126,126,-84,36,-9,1},{1,1,2,6,21,75,253,774,2130},40] (* _Harvey P. Dale_, Dec 24 2023 *)

%o (PARI) Vec((1 - 8*x + 29*x^2 - 60*x^3 + 81*x^4 - 66*x^5 + 40*x^6 - 15*x^7 + 3*x^8) / (1 - x)^9 + O(x^40)) \\ _Colin Barker_, Nov 11 2017

%K nonn,easy

%O 0,3

%A _R. J. Mathar_, Nov 08 2017