A075670 - OEIS

%I #21 Sep 08 2022 08:45:07

%S 1,20195,12292965,1561991824,77226633575,2014634387961,33098483802475,

%T 383318212734080,3377498614484589,23898971839102975,

%U 141290020118952881,719054471032657200,3223613105991831475,12964037775857022869

%N Sum of next n 9th powers.

%H Vincenzo Librandi, <a href="/A075670/b075670.txt">Table of n, a(n) for n = 1..10000</a>

%F a(1)=1; a(n)=sum(i^s, {i, n(n-1)/2+1, n(n-1)/2+1+n}).

%F a(n) = (5n^19 + 105n^17 + 666n^15 + 1530n^13 + 689n^11 - 995n^9 + 304n^7 + 640n^5 - 384n^3)/2560.

%F G.f.: x*(x^18 +20175*x^17 +11889255*x^16 +1319968434*x^15 +48299442990*x^14 +752964012192*x^13 +5757432094050*x^12 +23468751060270*x^11 +53583908362248*x^10 +70362713036770*x^9 +53583908362248*x^8 +23468751060270*x^7 +5757432094050*x^6+752964012192*x^5 +48299442990*x^4 +1319968434*x^3 +11889255*x^2 +20175*x +1)/(x -1)^20. [_Colin Barker_, Sep 06 2012]

%e s=9; a(1) = 1^s = 1; a(2) = 2^s + 3^s = 2^9 + 3^9 = 20195; a(3) = 4^s + 5^s + 6^s = 12292965, a(4) = 7^s + 8^s + 9^s + 10^3 = 1561991824.

%t i1 := n(n-1)/2+1; i2 := n(n-1)/2+n; s=9; Table[Sum[i^s, {i, i1, i2}], {n, 20}]

%t Total[#^9]&/@(Range[First[#]+1,Last[#]]&/@Partition[Accumulate[Range[ 0,15]],2,1]) (* _Harvey P. Dale_, Oct 05 2011 *)

%o (Magma) [(5*n^19 + 105*n^17 + 666*n^15 + 1530*n^13 + 689*n^11 - 995*n^9 + 304*n^7 + 640*n^5 - 384*n^3)/2560 : n in [1..20]]; // _Vincenzo Librandi_, Oct 06 2011

%Y Cf. A072474 (s=2), A075664 - A075670 (s=3-10), A075671 (s=n).

%Y Cf. A006003, A072474, A075664 - A075671.

%K nonn,easy

%O 1,2

%A _Zak Seidov_, Sep 24 2002

%E Formula from _Charles R Greathouse IV_, Sep 17 2009