A248158 Expansion of (1 - 2*x^2)/(1 + x)^3. Second column of Riordan triangle A248156.

1, -3, 4, -4, 3, -1, -2, 6, -11, 17, -24, 32, -41, 51, -62, 74, -87, 101, -116, 132, -149, 167, -186, 206, -227, 249, -272, 296, -321, 347, -374, 402, -431, 461, -492, 524, -557, 591, -626, 662, -699, 737, -776, 816, -857, 899, -942, 986

Offset: 0

Author

Wolfdieter Lang, Oct 05 2014

Keywords

• sign
• easy

Comments

This is the column k=1 sequence of the Riordan triangle A248156 without a leading zero.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (-3,-3,-1).

Crossrefs

Cf. A046691, A148157, A248156(n+1,1).

Programs

Magma

[(-1)^n*(2+5*n-n^2)/2: n in [0..60]]; // G. C. Greubel, May 30 2025

Mathematica

Table[(-1)^n*(2+5*n-n^2)/2, {n,0,60}] (* G. C. Greubel, May 30 2025 *)

Python

def A248158(n): return (-1)**n*(2+5*n-n**2)//2
print([A248158(n) for n in range(51)]) # G. C. Greubel, May 30 2025

Formula

O.g.f.: (1 - 2*x^2)/(1 + x)^3 = -2/(1 + x) + 4/(1 + x)^2 - 1/(1 + x)^3.

a(n) = (-1)^n*(4*(2*n+1) - (n+1)*(n+2))/2, n >= 0.

a(n) = -3*(a(n-1) + a(n-2)) - a(n-3), n >= 3 with a(0) = 1, a(1) = -3 and a(2) = 4.

From R. J. Mathar, Mar 13 2021: (Start)

a(n) = (-1)^(n+1)*A046691(n-5).

a(n) + a(n+1) = A248157(n+1). (End)

E.g.f.: (1/2)*(2 - 4*x - x^2)*exp(-x). - G. C. Greubel, May 30 2025