A141531 Inverse binomial transform of A001651.

1, 1, 1, -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

Offset: 0

Author

Paul Curtz, Aug 12 2008

Keywords

• sign

Links

• Index entries for linear recurrences with constant coefficients, signature (-2).

Crossrefs

Cf. A000079, A122803, A123344.

Programs

Mathematica

CoefficientList[Series[(1+3x+3x^2)/(1+2x),{x,0,40}],x] (* or *) Join[ {1,1},NestList[-2#&,1,38]] (* Harvey P. Dale, May 04 2012 *)
Join[{1, 1},LinearRecurrence[{-2},{1},32]] (* Ray Chandler, Aug 12 2015 *)

PARI

Vec((1 + 3*x + 3*x^2)/(1 + 2*x) + O(x^40)) \\ Andrew Howroyd, Nov 03 2018

Formula

a(n) = A123344(n+1), n > 0.

a(n) = (-2)^n/4 = (-1)^n*A000079(n-2), n > 1.

O.g.f.: (1 + 3*x + 3*x^2)/(1 + 2*x). - R. J. Mathar, Aug 27 2008

a(n) = -2*a(n-1) for n >= 3; a(0)=1, a(1)=1, a(2)=1. - Harvey P. Dale, May 04 2012

G.f.: x+1/Q(0) where Q(k) = 1 + x*(k+1)/(1 - 1/(1 - (k+1)/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, Sep 23 2012

G.f.: 1+x/U(0) where U(k) = 1 - x*(k+4) + x*(k+3)/U(k+1); (continued fraction). - Sergei N. Gladkovskii, Oct 11 2012

a(n) = A122803(n-2) for n >= 2. - Georg Fischer, Nov 03 2018

E.g.f.: (3/4) + (3/2)*x + (1/4)*exp(-2*x). - Alejandro J. Becerra Jr., Feb 15 2021

Extensions

Edited and extended by R. J. Mathar, Aug 28 2008