A084633 Inverse binomial transform of repeated odd numbers.

1, 0, 2, -4, 8, -16, 32, -64, 128, -256, 512, -1024, 2048, -4096, 8192, -16384, 32768, -65536, 131072, -262144, 524288, -1048576, 2097152, -4194304, 8388608, -16777216, 33554432, -67108864, 134217728, -268435456, 536870912, -1073741824, 2147483648, -4294967296

Offset: 0

Author

Paul Barry, Jun 05 2003

Keywords

• easy
• sign

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (-2).

Crossrefs

Cf. A034008.

Programs

Magma

[n le 1 select 1-n else (0^n + (-2)^n)/2: n in [0..40]]; // G. C. Greubel, Mar 18 2023

Mathematica

Join[{1,0}, NestList[-2#&,2,40]] (* Harvey P. Dale, Dec 28 2015 *)

SageMath

[(0^n + (-2)^n)/2 + int(n==1) for n in range(41)] # G. C. Greubel, Mar 18 2023

Formula

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

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

From Colin Barker, Jan 06 2013: (Start)

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

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

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

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