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A232604 a(n) = 2^n * Sum_{k=0..n} k^p*q^k, where p=3, q=-1/2.

%I A232604 #22 Mar 31 2021 20:25:18
%S A232604 0,-1,6,-15,34,-57,102,-139,234,-261,478,-375,978,-241,2262,1149,6394,
%T A232604 7875,21582,36305,80610,151959,314566,616965,1247754,2479883,4977342,
%U A232604 9935001,19891954,39759519,79546038,159062285
%N A232604 a(n) = 2^n * Sum_{k=0..n} k^p*q^k, where p=3, q=-1/2.
%C A232604 The factor 2^n (i.e., |1/q|^n) is present to make the values integers.
%C A232604 See also A232600 and references therein for integer values of q.
%C A232604 The same values with different signs are produced by a(n) = n^3 - 2*a(n). The signs are all positive until n = 15, with negative signs on values for all subsequent odd indices. - _Richard R. Forberg_, Feb 17 2014.
%H A232604 Stanislav Sykora, <a href="/A232604/b232604.txt">Table of n, a(n) for n = 0..1000</a>
%H A232604 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 A232604 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (-2,2,8,7,2).
%F A232604 a(n) = (2^(n+1) + (-1)^n*(9*n^3+18*n^2+6*n-2))/27.
%F A232604 G.f.: x*(1-4*x+x^2) / ( (2*x-1)*(1+x)^4 ). - _R. J. Mathar_, Nov 23 2014
%F A232604 E.g.f.: (1/27)*(2*exp(2*x) - (2 +33*x -45*x^2 +9*x^3)*exp(-x)). - _G. C. Greubel_, Mar 31 2021
%F A232604 a(n) = - 2*a(n-1) + 2*a(n-2) + 8*a(n-3) + 7*a(n-4) + 2*a(n-5). - _Wesley Ivan Hurt_, Mar 31 2021
%e A232604 a(3) = 2^3 * (0^3/2^0 - 1^3/2^1 + 2^3/2^2 - 3^3/2^3) = 0-4+16-27 = -15.
%p A232604 A232604:= n-> (2^(n+1) +(-1)^n*(9*n^3 +18*n^2 +6*n -2))/27; seq(A232604(n), n=0..30); # _G. C. Greubel_, Mar 31 2021
%t A232604 LinearRecurrence[{-2,2,8,7,2}, {0,-1,6,-15,34}, 35] (* _G. C. Greubel_, Mar 31 2021 *)
%o A232604 (PARI) a(n)=(2^(n+1)+(-1)^n*(9*n^3+18*n^2+6*n-2))/27;
%o A232604 (Magma) [(2^(n+1) + (-1)^n*(9*n^3 +18*n^2 +6*n -2))/27: n in [0..35]]; // _G. C. Greubel_, Mar 31 2021
%o A232604 (Sage) [(2^(n+1) + (-1)^n*(9*n^3 +18*n^2 +6*n -2))/27 for n in (0..35)] # _G. C. Greubel_, Mar 31 2021
%Y A232604 Cf. A001045 (p=0,q=-1/2), A053088 (p=1,q=-1/2), A232603 (p=2,q=-1/2), A000225 (p=0,q=1/2), A000295 and A125128 (p=1,q=1/2), A047520 (p=2,q=1/2), A213575 (p=3,q=1/2), A232599 (p=3,q=-1), A232600 (p=1,q=-2), A232601 (p=2,q=-2), A232602 (p=3,q=-2).
%K A232604 sign,easy
%O A232604 0,3
%A A232604 _Stanislav Sykora_, Nov 27 2013