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Expansion of g.f. (1 + 2*x) / (1 + 2*x + 4*x^2).
3

%I #27 Jul 15 2024 17:55:40

%S 1,0,-4,8,0,-32,64,0,-256,512,0,-2048,4096,0,-16384,32768,0,-131072,

%T 262144,0,-1048576,2097152,0,-8388608,16777216,0,-67108864,134217728,

%U 0,-536870912,1073741824,0,-4294967296,8589934592,0,-34359738368,68719476736,0,-274877906944,549755813888,0

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

%H Colin Barker, <a href="/A104538/b104538.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (-2,-4).

%F a(n) = 2^n*A057078(n), where A057078(n) = U(2n, 1/2), U(n, x) Chebyshev polynomial of second kind.

%F a(n) = b(n) + b(n+1) where b(n) = 0, 1, -1, -3, 11, -11, -21, 85, -85, ... is the inverse binomial transform of A287479(n). - _Paul Curtz_, Aug 05 2018

%F a(n) = (((-1+i*sqrt(3))^n*(-i+sqrt(3)) + (-1-i*sqrt(3))^n*(i+sqrt(3)))) / (2*sqrt(3)) where i=sqrt(-1). - _Colin Barker_, Aug 08 2018

%F E.g.f.: exp(-x)*(sqrt(3)*cos(sqrt(3)*x) + sin(sqrt(3)*x))/sqrt(3). - _Stefano Spezia_, Jul 15 2024

%t CoefficientList[Series[(1+2x)/(1+2x+4x^2),{x,0,50}],x] (* or *) LinearRecurrence[ {-2,-4},{1,0},50] (* _Harvey P. Dale_, Sep 18 2022 *)

%o (PARI) a(n) = 2^n*polchebyshev(2*n, 2, 1/2); \\ _Michel Marcus_, Aug 05 2018

%o (PARI) Vec((1 + 2*x) / (1 + 2*x + 4*x^2) + O(x^40)) \\ _Colin Barker_, Aug 08 2018

%Y Cf. A057078, A287479.

%K easy,sign

%O 0,3

%A _Paul Barry_, Mar 13 2005