%I #34 Feb 14 2024 04:59:23

%S 0,0,0,0,1,1,4,7,18,37,84,179,390,833,1784,3791,8042,16989,35804,

%T 75243,157774,330105,689344,1436935,2990386,6213781,12893604,26719267,

%U 55302678,114333617,236123784,487160639,1004147450,2067947213,4255199084,8749007451,17975233502

%N Expansion of x^4*(1-x-x^2)/((1+x)*(1-2*x)*(1-x-2*x^2)).

%H Vincenzo Librandi, <a href="/A219754/b219754.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. 17 (Lemma 4.5).

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,3,-4,-4).

%F G.f.: x^4*(1-x-x^2)/((1+x)^2*(1-2*x)^2).

%F a(n) = (2^(n-4)*(3*n+5)+(3*n-2)*(-1)^n)/27 for n=1 and n>2, a(0)=a(1)=a(2)=0. [_Bruno Berselli_, Nov 29 2012]

%t CoefficientList[Series[x^4 (1 - x - x^2)/((1 + x) (1 - 2 x) (1 - x - 2 x^2)), {x, 0, 36}], x] (* _Bruno Berselli_, Nov 30 2012 *)

%o (Maxima) makelist(coeff(taylor(x^4*(1-x-x^2)/((1+x)*(1-2*x)*(1-x-2*x^2)), x, 0, n), x, n), n, 0, 36); /* _Bruno Berselli_, Nov 29 2012 */

%o (Magma) I:=[0, 0, 0, 0, 1, 1, 4, 7]; [n le 8 select I[n] else 2*Self(n-1) + 3*Self(n-2) - 4*Self(n-3) - 4*Self(n-4): n in [1..40]]; // _Vincenzo Librandi_, Dec 15 2012

%Y Cf. A219751-A219759, A219837.

%K nonn,easy

%O 0,7

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