A248160 Expansion of (1 - 2*x^2)/(1 + x)^5. Fourth column of Riordan triangle A248156.

1, -5, 13, -25, 40, -56, 70, -78, 75, -55, 11, 65, -182, 350, -580, 884, -1275, 1767, -2375, 3115, -4004, 5060, -6302, 7750, -9425, 11349, -13545, 16037, -18850, 22010, -25544, 29480, -33847, 38675, -43995, 49839, -56240, 63232, -70850, 79130, -88109, 97825, -108317, 119625, -131790

Offset

0,2

Comments

This is column k=3 of the Riordan triangle A248156 without the leading three zeros.

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (-5,-10,-10,-5,-1).

Formula

O.g.f.: (1 - 2*x^2)/(1 + x)^5 = -2/(1 + x)^3 + 4/(1 + x)^4 - 1/(1 + x)^5.

a(n) = (-1)^n*(n+1)*(n+2)*(12 + 9*n - n^2)/4!.

a(n) = -5*(a(n-1) + a(n-4)) - 10*(a(n-2) + a(n-3)) - a(n-5), n >= 5, with a(0) =1, a(1) = -5, a(2) = 13, a(3) = -25 and a(4) = 40.

E.g.f.: (1/4!)*(24 - 96*x + 48*x^2 - x^4)*exp(-x). - G. C. Greubel, May 30 2025

Maple

A248160:=n->(-1)^n*(n+1)*(n+2)*(12 + 9*n - n^2)/4!: seq(A248160(n), n=0..30); # Wesley Ivan Hurt, Oct 09 2014

Mathematica

Table[(-1)^n*(n + 1)*(n + 2)*(12 + 9*n - n^2)/4!, {n, 0, 30}] (* Wesley Ivan Hurt, Oct 09 2014 *)
CoefficientList[Series[(1-2x^2)/(1+x)^5, {x, 0, 50}], x] (* or *) LinearRecurrence[ {-5, -10, -10, -5, -1}, {1, -5, 13, -25, 40}, 50] (* Harvey P. Dale, Apr 13 2019 *)

Prog

(PARI) Vec((1 - 2*x^2)/(1 + x)^5 + O(x^50)) \\ Michel Marcus, Oct 09 2014
(Magma)
[(-1)^n*(n + 1)*(n + 2)*(12 + 9*n - n^2)/24: n in [0..50]]; // G. C. Greubel, May 30 2025
(Python)
def A248160(n): return (-1)**n*(n+1)*(n+2)*(12+9*n-n**2)//24 # G. C. Greubel, May 30 2025

Crossrefs

Cf. A248156 (column k=3).

Cf. A248157 (k=0), A248158 (k=1), A248159 (k=2).

Sequence in context: A094079 A194811 A112558 * A098972 A081961 A096891

Adjacent sequences: A248157 A248158 A248159 * A248161 A248162 A248163

Keyword

sign,easy

Author

Wolfdieter Lang, Oct 09 2014

Status

approved