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%I #37 Mar 31 2021 22:28:28
%S 0,2,34,250,1274,5274,19098,63002,194074,567322,1591322,4317210,
%T 11395098,29392922,74350618,184942618,453378074,1097334810,2626158618,
%U 6222250010,14610858010,34032582682,78693531674,180757725210,412685959194
%N a(n) = 26 + 2^(n+1)*(-13 +9*n -3*n^2 +n^3).
%D M. Petkovsek et al., A=B, Peters, 1996, p. 97.
%H Reinhard Zumkeller, <a href="/A036827/b036827.txt">Table of n, a(n) for n = 0..1000</a>
%H 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 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (9,-32,56,-48,16).
%F a(n) = Sum_{k=0..n} 2^k*k^3. - _Benoit Cloitre_, Jun 11 2003
%F G.f.: 2*x*(1 +8*x +4*x^2)/((1-x)*(1-2*x)^4). - Maksym Voznyy (voznyy(AT)mail.ru), Jul 26 2009
%F a(n) = 9*a(n-1) -32*a(n-2) +56*a(n-3) -48*a(n-4) +16*a(n-5) for n>4 with a(0)=0, a(1)=2, a(2)=34, a(3)=250, a(4)=1274. - _Harvey P. Dale_, Dec 15 2011
%F a(n) = Sum_{k=0..n} Sum_{i=0..n} k^3 * C(k,i). - _Wesley Ivan Hurt_, Sep 21 2017
%F E.g.f.: 2 (13*exp(x) + (-13 +14*x +8*x^3)*exp(2*x)). - _G. C. Greubel_, Mar 31 2021
%e a(3) = 2^0*0^3 + 2^1*1^3 + 2^2*2^3 + 2^3*3^3 = 250.
%p A036827:= n-> 2*(13 + 2^n*(-13 +9*n -3*n^2 +n^3)); seq(A026827(n), n=0..30); # _G. C. Greubel_, Mar 31 2021
%t Table[26 +2^(n+1)(-13 +9n -3n^2 +n^3), {n, 0, 30}] (* or *) LinearRecurrence[ {9, -32, 56, -48, 16}, {0, 2, 34, 250, 1274}, 31] (* _Harvey P. Dale_, Dec 15 2011 *)
%o (Haskell)
%o a036827 n = 2^(n+1) * (n^3 - 3*n^2 + 9*n - 13) + 26
%o -- _Reinhard Zumkeller_, May 24 2012
%o (PARI) a(n)=26+2^(n+1)*(-13+9*n-3*n^2+n^3) \\ _Charles R Greathouse IV_, Oct 07 2015
%o (Magma) [2*(13 + 2^n*(-13 +9*n -3*n^2 +n^3)): n in [0..35]]; // _G. C. Greubel_, Mar 31 2021
%o (Sage) [2*(13 + 2^n*(-13 +9*n -3*n^2 +n^3)) for n in (0..35)] # _G. C. Greubel_, Mar 31 2021
%Y 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), this sequence (p=3,q=2), A077925 (p=0,q=-2), A232600 (p=1,q=-2), A232601 (p=2,q=-2), A232602 (p=3,q=-2), A232603 (p=2,q=-1/2), A232604 (p=3,q=-1/2).
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_
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