A075670 Sum of next n 9th powers.

1, 20195, 12292965, 1561991824, 77226633575, 2014634387961, 33098483802475, 383318212734080, 3377498614484589, 23898971839102975, 141290020118952881, 719054471032657200, 3223613105991831475, 12964037775857022869, 47453810583528962775, 159982264435790734336

Offset: 1

Author

Zak Seidov, Sep 24 2002

Keywords

• nonn
• easy

Examples

			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.
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..10000
• Index entries for linear recurrences with constant coefficients, signature (20, -190, 1140, -4845, 15504, -38760, 77520, -125970, 167960, -184756, 167960, -125970, 77520, -38760, 15504, -4845, 1140, -190, 20, -1).

Crossrefs

Cf. A001017 (9th powers).

Cf. A006003, A072474 (for squares), A075664 - A075671 (3rd to 10th powers), A069876 (n-th powers).

Programs

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

Mathematica

i1 := n(n-1)/2+1; i2 := n(n-1)/2+n; s=9; Table[Sum[i^s, {i, i1, i2}], {n, 20}]
Total[#^9]&/@(Range[First[#]+1,Last[#]]&/@Partition[Accumulate[Range[ 0,15]],2,1]) (* Harvey P. Dale, Oct 05 2011 *)
With[{nn=20},Total/@TakeList[Range[(nn(nn+1))/2]^9,Range[nn]]] (* Harvey P. Dale, Aug 05 2025 *)

Formula

a(n) = Sum_{i=n(n-1)/2+1..n(n-1)/2+n} i^9.

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

G.f.: x*(x^18 +20175*x^17 [phone]*x^16 [phone]*x^15 [phone]*x^14 [phone]*x^13 [phone]*x^12 [phone]*x^11 [phone]*x^10 [phone]*x^9 [phone]*x^8 [phone]*x^7 [phone]*x^6+752964012192*x^5 [phone]*x^4 [phone]*x^3 [phone]*x^2 +20175*x +1)/(x -1)^20. - Colin Barker, Sep 06 2012