A099570 Expansion of ((1+x)^2 - x^3)/(1+x)^2.

1, 0, 0, -1, 2, -3, 4, -5, 6, -7, 8, -9, 10, -11, 12, -13, 14, -15, 16, -17, 18, -19, 20, -21, 22, -23, 24, -25, 26, -27, 28, -29, 30, -31, 32, -33, 34, -35, 36, -37, 38, -39, 40, -41, 42, -43, 44, -45, 46, -47, 48, -49, 50, -51, 52, -53, 54, -55, 56, -57, 58, -59, 60, -61, 62, -63, 64, -65, 66, -67, 68, -69, 70, -71, 72, -73

Offset: 0

Author

Paul Barry, Oct 22 2004

Keywords

• easy
• sign

Comments

Row sums of number triangle A099569.

Links

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

Crossrefs

Cf. A038608, A099569.

Programs

Magma

[1,0] cat [(-1)^n*(n-2): n in [2..100]]; // G. C. Greubel, Jul 25 2022

Mathematica

Table[(-1)^n*(n-2) -Boole[n==1]+3*Boole[n==0], {n,0,100}] (* G. C. Greubel, Jul 25 2022 *)

SageMath

[(-1)^n*(n-2) -bool(n==1) +3*bool(n==0) for n in (0..100)] # G. C. Greubel, Jul 25 2022

Formula

a(n) = 2*0^n + (-1)^n*(n-2) - Sum_{k=0..n} k*binomial(n, k)*(-1)^(n-k).

a(n) = -2*a(n-1) - a(n-2), n>3, with a(0) = 1, a(1) = 0, a(2) = 0, a(3) = -1.

From G. C. Greubel, Jul 25 2022: (Start)

a(n) = (-1)^n*(n-2) - [n=1] + 3*[n=0].

a(n) = A038608(n-2), for n >= 2, with a(0) = 1, a(1) = 0.

E.g.f.: 3 - x - (x+2)*exp(-x). (End)