A104555 - OEIS

%I #25 Jan 01 2023 03:13:18

%S 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 0,117,126,0,-145,-155,0,176,187,0,-210,-222,0,247,260,0,-287,-301,0,

%U 330,345,0,-376,-392,0,425,442,0,-477,-495

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

%C Image of C(n+1,2) under the Riordan array (1, x*(1-x)).

%H G. C. Greubel, <a href="/A104555/b104555.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (3,-6,7,-6,3,-1).

%F 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 a(n) = Sum_{k=0..n} binomial(k, n-k)(-1)^(n-k)*k(k+1)/2.

%F a(n) = Sum_{k=0..floor(n/2)} binomial(n-k, k)(-1)^k*(n-k)(n-k+1)/2.

%F 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 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 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 CoefficientList[Series[x*(1-x)/(1-x+x^2)^3, {x,0,60}], x]  (* _Harvey P. Dale_, Apr 13 2011 *)

%o (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 (Sage)

%o def A104555_list(prec):

%o     P.<x> = PowerSeriesRing(ZZ, prec)

%o     return P( x*(1-x)/(1-x+x^2)^3 ).list()

%o A104555_list(60) # _G. C. Greubel_, Jan 01 2023

%Y Cf. A076118, A095130.

%K easy,sign

%O 0,3

%A _Paul Barry_, Mar 14 2005