A248157 Expansion of (1 - 2*x^2)/(1 + x)^2.

1, -2, 1, 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

Offset: 0

Author

Wolfdieter Lang, Oct 05 2014

Keywords

• sign
• easy
• less

Comments

First column of Riordan triangle A248156.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..5000
• Index entries for linear recurrences with constant coefficients, signature (-2,-1).

Crossrefs

Cf. A038608, A097141, A248156 (row k=0).

Programs

Magma

A248157:= func< n | n eq 0 select 1 else (-1)^n*(3-n) >; // G. C. Greubel, May 16 2025

Mathematica

CoefficientList[Series[(1-2x^2)/(1+x)^2,{x,0,60}],x] (* or *) LinearRecurrence[{-2,-1},{1,-2,1},60] (* Harvey P. Dale, Aug 25 2023 *)

PARI

Vec((1 - 2*x^2)/(1 + x)^2 + O(x^80)) \\ Michel Marcus, Oct 11 2014

SageMath

def A248157(n): return (-1)^n*(3-n) - 2*int(n==0) # G. C. Greubel, May 16 2025

Formula

O.g.f.: (1 - 2*x^2)/(1 + x)^2 = -2 + 4/(1+x) - 1/(1+x)^2.

a(n) = -2*delta(n,0) + (-1)^n*(3 - n), n >= 0, with Kronecker delta(n,0) = 1 if n=0 else 0.

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

a(n) = A038608(n-3) = A097141(n-1), n>=3.

E.g.f.: (3+x)*exp(-x) - 2. - G. C. Greubel, May 16 2025