%I #30 Sep 08 2022 08:44:49

%S 1114,3413,8476,18247,35414,63529,107128,171851,264562,393469,568244,

%T 800143,1102126,1488977,1977424,2586259,3336458,4251301,5356492,

%U 6680279,8253574,10110073,12286376,14822107,17760034,21146189,25029988,29464351,34505822,40214689

%N a(n) = n + (n+1)^2 + (n+2)^3 + (n+3)^4 + (n+4)^5.

%H Colin Barker, <a href="/A027622/b027622.txt">Table of n, a(n) for n = 0..1000</a>

%H Patrick De Geest, <a href="http://www.worldofnumbers.com/sumpower.htm">Palindromic Quasi_Under_Squares of the form n+(n+1)^2</a>

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (6,-15,20,-15,6,-1).

%F From _Colin Barker_, Dec 05 2016: (Start)

%F a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6) for n>5.

%F G.f.: (1114-3271*x+4708*x^2-3694*x^3+1522*x^4-259*x^5) / (1-x)^6.

%F (End)

%F E.g.f.: (1114 +2299*x +1382*x^2 +324*x^3 +31*x^4 +x^5)*exp(x). - _G. C. Greubel_, Aug 05 2022

%p seq( add((n+j)^(j+1), j=0..4), n=0..30); # _G. C. Greubel_, Aug 05 2022

%t Table[n +(n+1)^2 +(n+2)^3 +(n+3)^4 +(n+4)^5, {n, 0, 29}] (* _Alonso del Arte_, Nov 22 2016 *)

%t Table[ReleaseHold@ Total@ MapIndexed[#1^First@ #2 &, Rest@ FactorList[ Pochhammer[Hold@ n, 5]][[All, 1]]], {n, 0, 29}] (* or *)

%t CoefficientList[Series[(1114 -3271x +4708x^2 -3694x^3 +1522x^4 -259x^5)/(1-x)^6, {x, 0, 29}], x] (* _Michael De Vlieger_, Dec 05 2016 *)

%t Table[Total[Table[(n+k)^(k+1),{k,0,4}]],{n,0,30}] (* or *) LinearRecurrence[{6,-15,20,-15,6,-1}, {1114,3413,8476,18247,35414,63529}, 30] (* _Harvey P. Dale_, Aug 04 2022 *)

%o (Magma)[n+(n+1)^2+(n+2)^3+(n+3)^4+(n+4)^5: n in [0..30]]; // _Vincenzo Librandi_, Dec 28 2010

%o (PARI) Vec((1114-3271*x+4708*x^2-3694*x^3+1522*x^4-259*x^5) / (1-x)^6 + O(x^30)) \\ _Colin Barker_, Dec 05 2016

%o (SageMath) [sum((n+j)^(j+1) for j in (0..4)) for n in (0..30)] # _G. C. Greubel_, Aug 05 2022

%Y Cf. A000027, A027620, A027621, A028387.

%K nonn,easy

%O 0,1

%A _Patrick De Geest_