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

%I A248159 #22 May 30 2025 15:49:23
%S A248159 1,-4,8,-12,15,-16,14,-8,-3,20,-44,76,-117,168,-230,304,-391,492,-608,
%T A248159 740,-889,1056,-1242,1448,-1675,1924,-2196,2492,-2813,3160,-3534,3936,
%U A248159 -4367,4828,-5320,5844,-6401,6992,-7618,8280,-8979,9716
%N A248159 Expansion of (1 - 2*x^2)/(1 + x)^4. Third column of Riordan triangle A248156.
%C A248159 This is the column k=2 sequence of the Riordan triangle A248156 without the leading two zeros.
%H A248159 G. C. Greubel, <a href="/A248159/b248159.txt">Table of n, a(n) for n = 0..1000</a>
%H A248159 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (-4,-6,-4,-1).
%F A248159 O.g.f.: (1 - 2*x^2)/(1 + x)^4 = -1/(1 + x)^4 + 4/(1 + x)^3 -2/(1 + x)^2.
%F A248159 a(n) = (-1)^n*(n+1)*(6 + 7*n - n^2)/3!, n >= 0.
%F A248159 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.
%F A248159 a(n) + a(n+1) = A248158(n+1). - _R. J. Mathar_, Mar 13 2021
%F A248159 E.g.f.: (1/6)*(6 - 18*x + 3*x^2 + x^3)*exp(-x). - _G. C. Greubel_, May 30 2025
%p A248159 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
%t A248159 Table[(-1)^(n + 1)*(n + 1)*(n^2 - 7*n - 6)/3!, {n, 0, 50}] (* _Wesley Ivan Hurt_, Oct 07 2014 *)
%o A248159 (Magma) [(-1)^(n+1)*(n+1)*(n^2-7*n-6)/Factorial(3) : n in [0..50]]; // _Wesley Ivan Hurt_, Oct 07 2014
%o A248159 (Python)
%o A248159 def A248159(n): return (-1)**n*(n+1)*(6+7*n-n**2)//6 # _G. C. Greubel_, May 30 2025
%Y A248159 Cf. A248156 (k=2).
%Y A248159 Cf. A248157 (k=0), A248158 (k=1).
%K A248159 sign,easy
%O A248159 0,2
%A A248159 _Wolfdieter Lang_, Oct 07 2014