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A321632 Expansion of e.g.f. (1 + sin(x))/exp(x).

%I A321632 #34 Jul 21 2021 14:24:22
%S A321632 1,0,-1,1,1,-5,9,-9,1,15,-31,31,1,-65,129,-129,1,255,-511,511,1,-1025,
%T A321632 2049,-2049,1,4095,-8191,8191,1,-16385,32769,-32769,1,65535,-131071,
%U A321632 131071,1,-262145,524289,-524289,1,1048575,-2097151,2097151,1,-4194305,8388609,-8388609
%N A321632 Expansion of e.g.f. (1 + sin(x))/exp(x).
%C A321632 A140323(n) = |a(4*n-1)| = |a(4*n-2)|, A247281(n) = |a(4*n+1)|.
%C A321632 The absolute values of the coefficients of the expansion of the reciprocal of this function are listed in A186364.
%H A321632 Colin Barker, <a href="/A321632/b321632.txt">Table of n, a(n) for n = 0..1000</a>
%H A321632 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (-3,-4,-2).
%F A321632 a(4*k)   = 1;
%F A321632 a(4*k+1) = (-4)^k - 1;
%F A321632 a(4*k+2) = -2*a(4*k+1) - 1 = -2*(-4)^k + 1;
%F A321632 a(4*k+3) =  2*a(4*k+1) + 1 =  2*(-4)^k - 1.
%F A321632 From _Colin Barker_, Nov 16 2018: (Start)
%F A321632 G.f.: (1 + 3*x + 3*x^2) / ((1 + x)*(1 + 2*x + 2*x^2)).
%F A321632 a(n) = (-1)^n + i/2*((-1-i)^n - (-1+i)^n), where i=sqrt(-1).
%F A321632 a(n) = -3*a(n-1) - 4*a(n-2) - 2*a(n-3) for n>2. (End)
%p A321632 seq(factorial(n)*coeff(series((1+sin(x))/exp(x),x=0,48),x,n),n=0..47);
%t A321632 With[{nn=50},CoefficientList[Series[(1+Sin[x])/Exp[x],{x,0,nn}],x] Range[ 0,nn]!] (* or *) LinearRecurrence[{-3,-4,-2},{1,0,-1},50] (* _Harvey P. Dale_, Jul 21 2021 *)
%o A321632 (PARI) Vec((1 + 3*x + 3*x^2) / ((1 + x)*(1 + 2*x + 2*x^2)) + O(x^40)) \\ _Colin Barker_, Nov 16 2018
%Y A321632 Cf. A140323, A186364, A247281.
%K A321632 sign,easy
%O A321632 0,6
%A A321632 _Paolo P. Lava_, Nov 16 2018