%I #30 Feb 14 2024 06:43:17

%S 0,0,0,0,2,20,120,570,2355,8841,30906,102187,323053,984354,2908671,

%T 8375521,23594410,65237027,177520325,476515378,1264297431,3321423193,

%U 8653113914,22386784603,57586262493,147447786562,376173191919,957113924753,2430649701066

%N Expansion of x^4*(2-12*x+24*x^2-8*x^3-41*x^4+57*x^5-16*x^6)/((1-x)*(1-3*x+x^2)*(1-2*x)^6).

%H Vincenzo Librandi, <a href="/A219759/b219759.txt">Table of n, a(n) for n = 0..1000</a>

%H M. H. Albert, M. D. Atkinson and Robert Brignall, <a href="http://arxiv.org/abs/1206.3183">The enumeration of three pattern classes</a>, arXiv:1206.3183 [math.CO] (2012), p. 30.

%H <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (16,-112,449,-1132,1852,-1952,1264,-448,64).

%F G.f.: x^4*(2-12*x+24*x^2-8*x^3-41*x^4+57*x^5-16*x^6)/((1-x)*(1-3*x+x^2)*(1-2*x)^6).

%p A219759 := proc(n)

%p if n <= 1 then

%p 0;

%p else

%p 2^n*(881*n^2/24-14393*n/60+137+7*n^4/24-49*n^3/8+n^5/120) -384+ 64*A001906(n+2) ;

%p %/64 ;

%p end if;

%p end proc:

%p seq(A219759(n),n=0..20) ; # _R. J. Mathar_, Aug 19 2022

%t CoefficientList[Series[x^4 (2 - 12 x + 24 x^2 - 8 x^3 - 41 x^4 + 57 x^5 - 16 x^6)/((1 - x) (1 - 3 x + x^2) (1 - 2 x)^6), {x, 0, 28}], x] (* _Bruno Berselli_, Nov 30 2012 *)

%t LinearRecurrence[{16,-112,449,-1132,1852,-1952,1264,-448,64},{0,0,0,0,2,20,120,570,2355,8841,30906},40] (* _Harvey P. Dale_, Mar 01 2023 *)

%o (Maxima) makelist(coeff(taylor(x^4*(2-12*x+24*x^2-8*x^3-41*x^4+57*x^5-16*x^6)/((1-x)*(1-3*x+x^2)*(1-2*x)^6), x, 0, n), x, n), n, 0, 28); /* _Bruno Berselli_, Nov 29 2012 */

%o (Magma) I:=[0, 0, 0, 0, 2, 20, 120, 570, 2355, 8841, 30906, 102187, 323053]; [n le 11 select I[n] else 16*Self(n-1) -112*Self(n-2) + 449*Self(n-3) - 1132*Self(n-4) + 1852*Self(n-5) - 1952*Self(n-6) + 1264*Self(n-7) - 448*Self(n-8) + 64*Self(n-9): n in [1..30]]; // _Vincenzo Librandi_, Dec 14 2012

%Y Cf. A219751-A219758, A219837.

%K nonn,easy

%O 0,5

%A _N. J. A. Sloane_, Nov 28 2012