A248159 Expansion of (1 - 2*x^2)/(1 + x)^4. Third column of Riordan triangle A248156.

1, -4, 8, -12, 15, -16, 14, -8, -3, 20, -44, 76, -117, 168, -230, 304, -391, 492, -608, 740, -889, 1056, -1242, 1448, -1675, 1924, -2196, 2492, -2813, 3160, -3534, 3936, -4367, 4828, -5320, 5844, -6401, 6992, -7618, 8280, -8979, 9716

Offset: 0

Author

Wolfdieter Lang, Oct 07 2014

Keywords

• sign
• easy

Comments

This is the column k=2 sequence of the Riordan triangle A248156 without the leading two zeros.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (-4,-6,-4,-1).

Crossrefs

Cf. A248156 (k=2).

Cf. A248157 (k=0), A248158 (k=1).

Programs

Magma

[(-1)^(n+1)*(n+1)*(n^2-7*n-6)/Factorial(3) : n in [0..50]]; // Wesley Ivan Hurt, Oct 07 2014

Maple

A248159:=n->(-1)^(n+1)*(n+1)*(n^2-7*n-6)/3!: seq(A248159(n),n=0..50); # Wesley Ivan Hurt, Oct 07 2014

Mathematica

Table[(-1)^(n + 1)*(n + 1)*(n^2 - 7*n - 6)/3!, {n, 0, 50}] (* Wesley Ivan Hurt, Oct 07 2014 *)

Python

def A248159(n): return (-1)**n*(n+1)*(6+7*n-n**2)//6 # G. C. Greubel, May 30 2025

Formula

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

a(n) = (-1)^n*(n+1)*(6 + 7*n - n^2)/3!, n >= 0.

a(n) = -4*(a(n-1) + a(n-3)) - 6*a(n-2) - a(n-4), n >= 4, with a(0) =1, a(1) = -4, a(2) = 8 and a(3) = -12.

a(n) + a(n+1) = A248158(n+1). - R. J. Mathar, Mar 13 2021

E.g.f.: (1/6)*(6 - 18*x + 3*x^2 + x^3)*exp(-x). - G. C. Greubel, May 30 2025