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A104555 Expansion of x*(1 - x)/(1 - x + x^2)^3.

%I A104555 #25 Jan 01 2023 03:13:18
%S A104555 0,1,2,0,-5,-7,0,12,15,0,-22,-26,0,35,40,0,-51,-57,0,70,77,0,-92,-100,
%T A104555 0,117,126,0,-145,-155,0,176,187,0,-210,-222,0,247,260,0,-287,-301,0,
%U A104555 330,345,0,-376,-392,0,425,442,0,-477,-495
%N A104555 Expansion of x*(1 - x)/(1 - x + x^2)^3.
%C A104555 Image of C(n+1,2) under the Riordan array (1, x*(1-x)).
%H A104555 G. C. Greubel, <a href="/A104555/b104555.txt">Table of n, a(n) for n = 0..1000</a>
%H A104555 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (3,-6,7,-6,3,-1).
%F A104555 a(n) = 3*a(n-1) - 6*a(n-2) + 7*a(n-3) - 6*a(n-4) + 3*a(n-5) - a(n-6).
%F A104555 a(n) = Sum_{k=0..n} binomial(k, n-k)(-1)^(n-k)*k(k+1)/2.
%F A104555 a(n) = Sum_{k=0..floor(n/2)} binomial(n-k, k)(-1)^k*(n-k)(n-k+1)/2.
%F A104555 a(3*n) = 0, a(3*n-2) = n*(3*n - 1)/2, a(3*n-1) = n*(3*n + 1)/2. - _Ralf Stephan_, May 20 2007
%F A104555 a(n) = ((Sum_{k=1..n+1} k^5) mod (Sum_{k=1..n+1} k^3))/((n+1)*(n+2))*(-1)^floor((n mod 6)/4). - _Gary Detlefs_, Oct 31 2011
%p A104555 S:=(j,n)->sum(k^j,k=1..n):seq((S(5,n+1)mod S(3,n+1))/((n+1)*(n+2))*(-1)^floor((n mod 6)/4), n=1..40). # _Gary Detlefs_, Oct 31 2011
%t A104555 CoefficientList[Series[x*(1-x)/(1-x+x^2)^3, {x,0,60}], x]  (* _Harvey P. Dale_, Apr 13 2011 *)
%o A104555 (Magma) R<x>:=PowerSeriesRing(Integers(), 60); [0] cat Coefficients(R!( x*(1-x)/(1-x+x^2)^3 )); // _G. C. Greubel_, Jan 01 2023
%o A104555 (Sage)
%o A104555 def A104555_list(prec):
%o A104555     P.<x> = PowerSeriesRing(ZZ, prec)
%o A104555     return P( x*(1-x)/(1-x+x^2)^3 ).list()
%o A104555 A104555_list(60) # _G. C. Greubel_, Jan 01 2023
%Y A104555 Cf. A076118, A095130.
%K A104555 easy,sign
%O A104555 0,3
%A A104555 _Paul Barry_, Mar 14 2005