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A061224 a(n) = n^2 + (n + 1)^3 + (n + 2)^4 + (n + 3)^5.

%I A061224 #26 Nov 14 2022 17:42:15
%S A061224 260,1114,3412,8474,18244,35410,63524,107122,171844,264554,393460,
%T A061224 568234,800132,1102114,1488964,1977410,2586244,3336442,4251284,
%U A061224 5356474,6680260,8253554,10110052,12286354,14822084,17760010,21146164,25029962
%N A061224 a(n) = n^2 + (n + 1)^3 + (n + 2)^4 + (n + 3)^5.
%H A061224 Vincenzo Librandi, <a href="/A061224/b061224.txt">Table of n, a(n) for n = 0..10000</a>
%H A061224 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (6,-15,20,-15,6,-1).
%F A061224 G.f.: 2*(130 - 223*x + 314*x^2 - 244*x^3 + 100*x^4 - 17*x^5)/(1-x)^6. - _Bruno Berselli_, Aug 05 2011
%F A061224 E.g.f.: exp(x)*(260 + 854*x + 722*x^2 + 220*x^3 + 26*x^4 + x^5). - _Stefano Spezia_, Nov 02 2018
%e A061224 a(1) = 1 + 2^3 + 3^4 + 4^5 = 1 + 8 + 81 + 1024 = 1114.
%p A061224 seq(n^2+(n+1)^3+(n+2)^4+(n+3)^5,n=0..30); # _Muniru A Asiru_, Nov 02 2018
%t A061224 Table[n^2 +(n+1)^3 +(n+2)^4 +(n+3)^5, {n,0,30}] (* _G. C. Greubel_, Nov 02 2018 *)
%t A061224 CoefficientList[Series[E^x (260 + 854 x + 722 x^2 + 220 x^3 + 26 x^4 + x^5), {x, 0, 50}], x]*Table[k!, {k, 0, 50}] (* _Stefano Spezia_, Nov 02 2018 *)
%t A061224 Table[260+440 n+298 n^2+99 n^3+16 n^4+n^5,{n,0,30}] (* or *) LinearRecurrence[ {6,-15,20,-15,6,-1},{260,1114,3412,8474,18244,35410},30] (* _Harvey P. Dale_, Nov 14 2022 *)
%o A061224 (Magma) [n^2 + (n + 1)^3 + (n + 2)^4 + (n + 3)^5: n in [0..30]]; // _Vincenzo Librandi_, Aug 05 2011
%o A061224 (PARI) vector(30, n, n--; n^2 +(n+1)^3 +(n+2)^4 +(n+3)^5) \\ _G. C. Greubel_, Nov 02 2018
%o A061224 (GAP) List([0..30],n->n^2+(n+1)^3+(n+2)^4+(n+3)^5); # _Muniru A Asiru_, Nov 02 2018
%Y A061224 Cf. A027621.
%K A061224 nonn,easy
%O A061224 0,1
%A A061224 _Olivier Gérard_, May 31 2001
%E A061224 Offset changed from 1 to 0 by _Vincenzo Librandi_, Aug 05 2011