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A075670 Sum of next n 9th powers.

%I A075670 #36 Aug 05 2025 18:06:09
%S A075670 1,20195,12292965,1561991824,77226633575,2014634387961,33098483802475,
%T A075670 383318212734080,3377498614484589,23898971839102975,
%U A075670 141290020118952881,719054471032657200,3223613105991831475,12964037775857022869,47453810583528962775,159982264435790734336
%N A075670 Sum of next n 9th powers.
%H A075670 Vincenzo Librandi, <a href="/A075670/b075670.txt">Table of n, a(n) for n = 1..10000</a>
%H A075670 <a href="/index/Rec#order_20">Index entries for linear recurrences with constant coefficients</a>, signature (20, -190, 1140, -4845, 15504, -38760, 77520, -125970, 167960, -184756, 167960, -125970, 77520, -38760, 15504, -4845, 1140, -190, 20, -1).
%F A075670 a(n) = Sum_{i=n(n-1)/2+1..n(n-1)/2+n} i^9.
%F A075670 a(n) = (5n^19 + 105n^17 + 666n^15 + 1530n^13 + 689n^11 - 995n^9 + 304n^7 + 640n^5 - 384n^3)/2560. - _Charles R Greathouse IV_, Sep 17 2009
%F A075670 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 A075670 a(1) = 1^9 = 1; a(2) = 2^9 + 3^9 = 20195; a(3) = 4^9 + 5^9 + 6^9 = 12292965; a(4) = 7^9 + 8^9 + 9^9 + 10^9 = 1561991824.
%t A075670 i1 := n(n-1)/2+1; i2 := n(n-1)/2+n; s=9; Table[Sum[i^s, {i, i1, i2}], {n, 20}]
%t A075670 Total[#^9]&/@(Range[First[#]+1,Last[#]]&/@Partition[Accumulate[Range[ 0,15]],2,1]) (* _Harvey P. Dale_, Oct 05 2011 *)
%t A075670 With[{nn=20},Total/@TakeList[Range[(nn(nn+1))/2]^9,Range[nn]]] (* _Harvey P. Dale_, Aug 05 2025 *)
%o A075670 (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 A075670 Cf. A001017 (9th powers).
%Y A075670 Cf. A006003, A072474 (for squares), A075664 - A075671 (3rd to 10th powers), A069876 (n-th powers).
%K A075670 nonn,easy
%O A075670 1,2
%A A075670 _Zak Seidov_, Sep 24 2002