This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A248160 #24 May 30 2025 23:52:58
%S A248160 1,-5,13,-25,40,-56,70,-78,75,-55,11,65,-182,350,-580,884,-1275,1767,
%T A248160 -2375,3115,-4004,5060,-6302,7750,-9425,11349,-13545,16037,-18850,
%U A248160 22010,-25544,29480,-33847,38675,-43995,49839,-56240,63232,-70850,79130,-88109,97825,-108317,119625,-131790
%N A248160 Expansion of (1 - 2*x^2)/(1 + x)^5. Fourth column of Riordan triangle A248156.
%C A248160 This is column k=3 of the Riordan triangle A248156 without the leading three zeros.
%H A248160 G. C. Greubel, <a href="/A248160/b248160.txt">Table of n, a(n) for n = 0..1000</a>
%H A248160 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (-5,-10,-10,-5,-1).
%F A248160 O.g.f.: (1 - 2*x^2)/(1 + x)^5 = -2/(1 + x)^3 + 4/(1 + x)^4 - 1/(1 + x)^5.
%F A248160 a(n) = (-1)^n*(n+1)*(n+2)*(12 + 9*n - n^2)/4!.
%F A248160 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.
%F A248160 E.g.f.: (1/4!)*(24 - 96*x + 48*x^2 - x^4)*exp(-x). - _G. C. Greubel_, May 30 2025
%p A248160 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
%t A248160 Table[(-1)^n*(n + 1)*(n + 2)*(12 + 9*n - n^2)/4!, {n, 0, 30}] (* _Wesley Ivan Hurt_, Oct 09 2014 *)
%t A248160 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 *)
%o A248160 (PARI) Vec((1 - 2*x^2)/(1 + x)^5 + O(x^50)) \\ _Michel Marcus_, Oct 09 2014
%o A248160 (Magma)
%o A248160 [(-1)^n*(n + 1)*(n + 2)*(12 + 9*n - n^2)/24: n in [0..50]]; // _G. C. Greubel_, May 30 2025
%o A248160 (Python)
%o A248160 def A248160(n): return (-1)**n*(n+1)*(n+2)*(12+9*n-n**2)//24 # _G. C. Greubel_, May 30 2025
%Y A248160 Cf. A248156 (column k=3).
%Y A248160 Cf. A248157 (k=0), A248158 (k=1), A248159 (k=2).
%K A248160 sign,easy
%O A248160 0,2
%A A248160 _Wolfdieter Lang_, Oct 09 2014