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