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Inverse binary transform of A027656.
1

%I #40 Apr 24 2023 09:34:49

%S 1,-1,3,-7,16,-36,80,-176,384,-832,1792,-3840,8192,-17408,36864,

%T -77824,163840,-344064,720896,-1507328,3145728,-6553600,13631488,

%U -28311552,58720256,-121634816,251658240,-520093696,1073741824,-2214592512,4563402752,-9395240960,19327352832

%N Inverse binary transform of A027656.

%H Vincenzo Librandi and Evert Provoost, <a href="/A081037/b081037.txt">Table of n, a(n) for n = 0..1000</a> [Terms 0 through 300 were computed by Vincenzo Librandi; terms 301 through 1000 by Evert Provoost, Jan 25 2016]

%H F. Ellermann, <a href="/A001792/a001792.txt">Illustration of binomial transforms</a>

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

%F G.f.: (1+x)^3/(1+2*x)^2.

%F a(n) = (-1)^n*A045623(n+1)/4, n>1. - _R. J. Mathar_, Sep 27 2012

%F Recurrence: a(n) = -4a(n-1) - 4a(n-2), a(0)=1, a(1)=-1, a(2)=3, a(3)=-7. - _Ralf Stephan_, Jul 14 2013

%F E.g.f.: exp(-x)*(2*cosh(x) + x*sinh(x))/2. - _Stefano Spezia_, Apr 24 2023

%t CoefficientList[Series[(1 + x)^3 / (1 + 2 x)^2, {x, 0, 30}], x] (* _Vincenzo Librandi_, Aug 06 2013 *)

%t LinearRecurrence[{-4,-4},{1,-1,3,-7},40] (* _Harvey P. Dale_, Apr 14 2019 *)

%o (Magma) I:=[1, -1, 3, -7]; [n le 4 select I[n] else -4*Self(n-1)-4*Self(n-2): n in [1..40]]; // _Vincenzo Librandi_, Aug 06 2013

%o (PARI) Vec((1+x)^3/(1+2*x)^2 + O(x^40)) \\ _Michel Marcus_, Jan 25 2016

%Y Cf. A027656, A045623, A045891.

%K easy,sign

%O 0,3

%A _Paul Barry_, Mar 03 2003