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Inverse binomial transform of repeated odd numbers.
6

%I #21 Mar 19 2023 02:48:26

%S 1,0,2,-4,8,-16,32,-64,128,-256,512,-1024,2048,-4096,8192,-16384,

%T 32768,-65536,131072,-262144,524288,-1048576,2097152,-4194304,8388608,

%U -16777216,33554432,-67108864,134217728,-268435456,536870912,-1073741824,2147483648,-4294967296

%N Inverse binomial transform of repeated odd numbers.

%H G. C. Greubel, <a href="/A084633/b084633.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_01">Index entries for linear recurrences with constant coefficients</a>, signature (-2).

%F a(n) = (0^n + (-2)^n)/2, for n > 1, with a(1) = 0.

%F abs(a(n)) = A034008(n).

%F From _Colin Barker_, Jan 06 2013: (Start)

%F a(n) = (-1)^n * 2^(n-1) for n > 1.

%F a(n) = -2*a(n-1) for n > 2.

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

%F E.g.f.: (1 + 2*x + exp(-2*x))/2. - _Alejandro J. Becerra Jr._, Jan 29 2021

%t Join[{1,0}, NestList[-2#&,2,40]] (* _Harvey P. Dale_, Dec 28 2015 *)

%o (Magma) [n le 1 select 1-n else (0^n + (-2)^n)/2: n in [0..40]]; // _G. C. Greubel_, Mar 18 2023

%o (SageMath) [(0^n + (-2)^n)/2 + int(n==1) for n in range(41)] # _G. C. Greubel_, Mar 18 2023

%Y Cf. A034008.

%K easy,sign

%O 0,3

%A _Paul Barry_, Jun 05 2003