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A232600 a(n) = Sum_{k=0..n} k^p*q^k, where p=1, q=-2.

%I A232600 #35 Mar 31 2021 20:25:00
%S A232600 0,-2,6,-18,46,-114,270,-626,1422,-3186,7054,-15474,33678,-72818,
%T A232600 156558,-334962,713614,-1514610,3203982,-6757490,14214030,-29826162,
%U A232600 62448526,-130489458,272163726,-566697074,1178133390,-2445745266,5070447502,-10498808946,21713445774
%N A232600 a(n) = Sum_{k=0..n} k^p*q^k, where p=1, q=-2.
%H A232600 Stanislav Sykora, <a href="/A232600/b232600.txt">Table of n, a(n) for n = 0..1000</a>
%H A232600 S. Sykora, <a href="http://dx.doi.org/10.3247/SL1Math06.002">Finite and Infinite Sums of the Power Series (k^p)(x^k)</a>, DOI 10.3247/SL1Math06.002, Section V.
%H A232600 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (-3,0,4).
%F A232600 a(n) = 2*( (3*n+1)*(-2)^n - 1 )/9.
%F A232600 abs(a(n)) = 2*A045883(n) = A140960(n).
%F A232600 From _Bruno Berselli_, Nov 28 2013: (Start)
%F A232600 G.f.: -2*x / ((1 - x)*(1 + 2*x)^2). [corrected by _Georg Fischer_, May 11 2019]
%F A232600 a(n) = -3*a(n-1) +4*a(n-3). (End)
%F A232600 From _G. C. Greubel_, Mar 31 2021: (Start)
%F A232600 E.g.f.: (2/9)*(-exp(x) + (1-6*x)*exp(-2*x)).
%F A232600 a(n) = 2*(-1)^n*A045883(n). (End)
%e A232600 a(3) = 0^1*2^0 - 1^1*2^1 + 2^1*2^2 - 3^1*2^3 = -18.
%p A232600 A232600:= n-> 2*((-2)^n*(3*n+1) -1)/9; seq(A232600(n), n=0..30); # _G. C. Greubel_, Mar 31 2021
%t A232600 Table[2((3n+1)(-2)^n -1)/9, {n, 0, 30}] (* _Bruno Berselli_, Nov 28 2013 *)
%o A232600 (PARI) a(n)=-((3*n+1)*(-2)^(n+1)+2)/9;
%o A232600 (Magma) [2*((-2)^n*(3*n+1) -1)/9: n in [0..30]]; // _G. C. Greubel_, Mar 31 2021
%o A232600 (Sage) [2*((-2)^n*(3*n+1) -1)/9 for n in (0..30)] # _G. C. Greubel_, Mar 31 2021
%Y A232600 Cf. A045883, A140960 (absolute values), A059841 (p=0, q=-1), A130472 (p=1 ,q=-1), A089594 (p=2, q=-1), A232599 (p=3, q=-1), A126646 (p=0, q=2), A036799 (p=1, q=2), A036800 (p=q=2), A036827 (p=3, q=2), A077925 (p=0, q=-2), A232601 (p=2, q=-2), A232602 (p=3, q=-2), A232603 (p=2, q=-1/2), A232604 (p=3, q=-1/2).
%Y A232600 Cf. A045883.
%K A232600 sign,easy
%O A232600 0,2
%A A232600 _Stanislav Sykora_, Nov 27 2013