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A232601 a(n) = Sum_{k=0..n} k^p*q^k for p = 2 and q = -2.

%I A232601 #30 Mar 31 2021 17:25:31
%S A232601 0,-2,14,-58,198,-602,1702,-4570,11814,-29658,72742,-175066,414758,
%T A232601 -969690,2241574,-5131226,11645990,-26233818,58700838,-130567130,
%U A232601 288863270,-635980762,1394062374,-3043511258,6620165158
%N A232601 a(n) = Sum_{k=0..n} k^p*q^k for p = 2 and q = -2.
%H A232601 Stanislav Sykora, <a href="/A232601/b232601.txt">Table of n, a(n) for n = 0..1000</a>
%H A232601 Stanislav 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 A232601 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (-5,-6,4,8).
%F A232601 a(n) = 2*((-2)^n * (9*n^2 + 6*n - 1) + 1)/27.
%F A232601 G.f.: 2*x*(-1 + 2*x) / ((1-x)*(1+2*x)^3). - _R. J. Mathar_, Nov 23 2014
%F A232601 E.g.f.: (2/27)*(exp(x) - (1 +30*x -36*x^2)*exp(-2*x)). - _G. C. Greubel_, Mar 31 2021
%F A232601 a(n) = - 5*a(n-1) - 6*a(n-2) + 4*a(n-3) + 8*a(n-4). - _Wesley Ivan Hurt_, Mar 31 2021
%e A232601 a(3) = 0^2*2^0 - 1^2*2^1 + 2^2*2^2 - 3^2*2^3 = -58.
%p A232601 A232601:= n-> 2*(1 - (-2)^n*(1-6*n-9*n^2))/27; seq(A232601(n), n=0..30); # _G. C. Greubel_, Mar 31 2021
%t A232601 LinearRecurrence[{-5,-6,4,8},{0,-2,14,-58},30] (* _Harvey P. Dale_, Aug 20 2015 *)
%o A232601 (PARI) S2M2(n)=((-1)^n*2^(n+1)*(9*n^2+6*n-1)+2)/27;
%o A232601 v = vector(10001); for(k=1, #v, v[k]=S2M2(k-1))
%o A232601 (Magma) [2*(1 - (-2)^n*(1-6*n-9*n^2))/27: n in [0..30]]; // _G. C. Greubel_, Mar 31 2021
%o A232601 (Sage) [2*(1 - (-2)^n*(1-6*n-9*n^2))/27 for n in (0..30)] # _G. C. Greubel_, Mar 31 2021
%Y A232601 Cf. 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=2,q=2), A036827 (p=3,q=2), A077925 (p=0,q=-2), A232600 (p=1,q=-2), A232602 (p=3,q=-2), A232603 (p=2,q=-1/2), A232604 (p=3,q=-1/2).
%K A232601 sign,easy
%O A232601 0,2
%A A232601 _Stanislav Sykora_, Nov 27 2013