A171501 Inverse binomial transform of A084640.

0, 1, 3, -1, 7, -9, 23, -41, 87, -169, 343, -681, 1367, -2729, 5463, -10921, 21847, -43689, 87383, -174761, 349527, -699049, 1398103, -2796201, 5592407, -11184809, 22369623, -44739241, 89478487, -178956969, 357913943, -715827881

Offset: 0

Author

Paul Curtz, Dec 10 2009

Keywords

• easy
• sign

Comments

a(n) and differences are

0, 1, 3, -1, 7, -9,

1, 2, -4, 8, -16, 32, =(-1)^(n+1) * A171449(n),

1, -6, 12, -24, 48, -96,

-7, 18, -36, 72, -144, 288,

25, -54, 108, -216, 432, -864,

Vertical: 1) 0 followed with A168589(n).

2) (-1 followed with A008776(n) ) signed. See A025192(n).

Main diagonal: see A167747(1+n). - Paul Curtz, Jun 16 2011

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (-1,2).

Programs

Magma

I:=[0, 1, 3]; [n le 3 select I[n] else -Self(n-1) + 2*Self(n-2): n in [1..40]]; // Vincenzo Librandi, Oct 18 2012

Mathematica

CoefficientList[Series[x*(1 + 4*x)/((1 + 2*x)*(1 - x)), {x, 0, 30}], x]
LinearRecurrence[{-1,2},{0,1,3},40] (* Harvey P. Dale, Jan 14 2020 *)

Formula

a(n) = A140966(n), n>0.

G.f.: x*(1+4*x) / ( (1+2*x)*(1-x) ). - R. J. Mathar, Jun 14 2011

a(1+n)= (-1)^(1+n) * A001045(1+n) + 2. - Paul Curtz, Jun 16 2011

Extensions

Mathematica program by Olivier Gérard, Jul 06 2011